In Chapter 10 the authors talk about three phases of learning facts: counting strategies, reasoning strategies, and mastery. The advantages of an approach that recognizes this developmental progression include:
1) Studies have shown that memorization simple doesn't work. It was interesting to read that students, on their own, develop strategies to explain what they are memorizing. What's more, memorization works against fluency, pigeon-holing students into a set way of doing something rather than allowing them to choose from a variety of strategies. 2. Learning facts is a progression, with one phase supporting the next. Often, however, the reasoning phase is skipped; without reasoning, students fluency is hindered; they can not choose flexibly between strategy options. Reasoning strategies for addition, for example, are directly related to one or more number relationships. Strategies learned in K-1 are foundational to later strategies. To support the progression of learning, reasoning must be taught explicitly.
"This developmental progression takes time and many [student] experiences." "Don't expect to have a strategy introduced and understood with just one lesson or activity."
What's more, fluency with addition and subtraction is critical when learning strategies for multiplication. Without a developmental progression that builds fluency at each level, students struggle.
I agree that it was very interesting to read that students develop their own strategies to explain what they are memorizing. I think students are capable of a lot more than we still even give them credit for. I think it can be so easy to skip over the reasoning phase when a teacher believes a student understands the math. With the best of intentions, teachers may push forward without really ensuring students have developed the strong number fluency.
The ideas from this chapter that made most sense to me are reasoning and developmental progression. Dean highlights these in his post above. Reasoning seems so logical and is the "making sense of the numbers" part of math. Students who understand the reasoning, gain the flexibility and number sense Jo Boaler talks about too. Developmental Progression makes sense because it is all about logical sequencing and the use of meta cognition to teach a concept. This chapter was full of useful concepts.
I agree with Dean and think that it is important for teachers to give students lots and lots of time to practice their new strategies in engaging ways, like playing games. I like how the text mentions that students should not be rushed into the next learning goal without proper time to practice and master the current strategies and skills, as well. All of the stages of number sense are important, as knowledge builds on knowledge. The methods in this chapter prioritize student learning above all else, which is how it should be.
In reading the chapter combined with the other article and my current classroom experience I am thinking that the unfortunate stress on building the knowledge has been overshadowed by people recognizing that for a portion of people the memorization of facts does not work. As Jo says in her article that it is important that the facts are learned or that proficiency is reached because the use of the ability reoccurs in mathematics over and over again. So I believe the standard should be stressed that proficiency with the calculation skills is paramount and not something that students should be allowed to not gain and keep moving on. Currently when the 6th graders I am working with fail to have the numeration skills to complete basic calculations the conceptual work becomes guess work with no way to properly check their work. It use to be that proofs would help a student understand their work and confirm their answer. But I am finding that many of my current students are not able to estimate to ball park check and to complete calculations to check their work. The activities are definitely a great tool to have to help teachers not only understand that there are other ways to teach the number facts but to truly have the methods at hand to teach the skills needed. I love the idea of using clocks to teach multiplication. Students could learn two skills at once! The bowling game is another example of fun reinforcement for the calculation skills. But truly I feel to many students have gotten a message that they will be able to simply use a calculator and often do not have the number sense to understand if their answer is correct. Calculators magnify problems for many of my students. Meg's last line about the methods in the chapter prioritize student learning above all is the message and if we use different methods for each of our students to arrive at the point of proficiency we win.
I feel similar that the focus on number sense and understanding the math involved has been overlooked at some point due to a focus on memorization. The use of tools and pushing students along to new concepts without them understanding number relationships has led them to solely rely on counting strategies to solve problems. For some students it seems necessary to take a step back and develop their number sense before being able to move forward.
This chapter describes disadvantages of memorization of math facts, as well as advantages of the developmental approach. The main disadvantage of memorization of math facts, is that is does not work for most people. For the few that can memorize math facts in the traditional way, the result is not increased understanding of math. For the majority who struggle to memorize, the result is anxiety and a feeling of failure. My own (vivid!) memory of counting by ones on my fingers behind my back while I recited my multiplication tables in front of the class in third grade is evidence of this! In contrast, the developmental approach helps students achieve fluency by focusing on building number sense - helping students to recognize patterns and use strategies to solve problems. This approach builds student confidence in math ability by supporting student thinking and encouraging students to use what they know to find facts that they do not yet know. The developmental approach encourages flexibility in problem solving, helping them see problems and solutions from different points of view. Building number sense in these ways helps students build the foundation they will need as they advance through math, and also allows more students to be successful with numbers from a young age.
In Chapter 10 the authors talk about three phases of learning facts: counting strategies, reasoning strategies, and mastery. The advantages of an approach that recognizes this developmental progression include:
ReplyDelete1) Studies have shown that memorization simple doesn't work. It was interesting to read that students, on their own, develop strategies to explain what they are memorizing. What's more, memorization works against fluency, pigeon-holing students into a set way of doing something rather than allowing them to choose from a variety of strategies.
2. Learning facts is a progression, with one phase supporting the next. Often, however, the reasoning phase is skipped; without reasoning, students fluency is hindered; they can not choose flexibly between strategy options. Reasoning strategies for addition, for example, are directly related to one or more number relationships. Strategies learned in K-1 are foundational to later strategies. To support the progression of learning, reasoning must be taught explicitly.
"This developmental progression takes time and many [student] experiences." "Don't expect to have a strategy introduced and understood with just one lesson or activity."
What's more, fluency with addition and subtraction is critical when learning strategies for multiplication. Without a developmental progression that builds fluency at each level, students struggle.
I agree that it was very interesting to read that students develop their own strategies to explain what they are memorizing. I think students are capable of a lot more than we still even give them credit for. I think it can be so easy to skip over the reasoning phase when a teacher believes a student understands the math. With the best of intentions, teachers may push forward without really ensuring students have developed the strong number fluency.
DeleteThe ideas from this chapter that made most sense to me are reasoning and developmental progression. Dean highlights these in his post above. Reasoning seems so logical and is the "making sense of the numbers" part of math. Students who understand the reasoning, gain the flexibility and number sense Jo Boaler talks about too. Developmental Progression makes sense because it is all about logical sequencing and the use of meta cognition to teach a concept. This chapter was full of useful concepts.
DeleteI agree with Dean and think that it is important for teachers to give students lots and lots of time to practice their new strategies in engaging ways, like playing games. I like how the text mentions that students should not be rushed into the next learning goal without proper time to practice and master the current strategies and skills, as well. All of the stages of number sense are important, as knowledge builds on knowledge. The methods in this chapter prioritize student learning above all else, which is how it should be.
ReplyDeleteIn reading the chapter combined with the other article and my current classroom experience I am thinking that the unfortunate stress on building the knowledge has been overshadowed by people recognizing that for a portion of people the memorization of facts does not work. As Jo says in her article that it is important that the facts are learned or that proficiency is reached because the use of the ability reoccurs in mathematics over and over again. So I believe the standard should be stressed that proficiency with the calculation skills is paramount and not something that students should be allowed to not gain and keep moving on. Currently when the 6th graders I am working with fail to have the numeration skills to complete basic calculations the conceptual work becomes guess work with no way to properly check their work. It use to be that proofs would help a student understand their work and confirm their answer. But I am finding that many of my current students are not able to estimate to ball park check and to complete calculations to check their work. The activities are definitely a great tool to have to help teachers not only understand that there are other ways to teach the number facts but to truly have the methods at hand to teach the skills needed. I love the idea of using clocks to teach multiplication. Students could learn two skills at once! The bowling game is another example of fun reinforcement for the calculation skills. But truly I feel to many students have gotten a message that they will be able to simply use a calculator and often do not have the number sense to understand if their answer is correct. Calculators magnify problems for many of my students. Meg's last line about the methods in the chapter prioritize student learning above all is the message and if we use different methods for each of our students to arrive at the point of proficiency we win.
ReplyDeleteI feel similar that the focus on number sense and understanding the math involved has been overlooked at some point due to a focus on memorization. The use of tools and pushing students along to new concepts without them understanding number relationships has led them to solely rely on counting strategies to solve problems. For some students it seems necessary to take a step back and develop their number sense before being able to move forward.
DeleteThis chapter describes disadvantages of memorization of math facts, as well as advantages of the developmental approach. The main disadvantage of memorization of math facts, is that is does not work for most people. For the few that can memorize math facts in the traditional way, the result is not increased understanding of math. For the majority who struggle to memorize, the result is anxiety and a feeling of failure. My own (vivid!) memory of counting by ones on my fingers behind my back while I recited my multiplication tables in front of the class in third grade is evidence of this! In contrast, the developmental approach helps students achieve fluency by focusing on building number sense - helping students to recognize patterns and use strategies to solve problems. This approach builds student confidence in math ability by supporting student thinking and encouraging students to use what they know to find facts that they do not yet know. The developmental approach encourages flexibility in problem solving, helping them see problems and solutions from different points of view. Building number sense in these ways helps students build the foundation they will need as they advance through math, and also allows more students to be successful with numbers from a young age.
ReplyDelete