Reflect:
After reading Chapter 4, please reflect on the questions below and post your response by Monday. Feel free to respond to any of the questions provided or share something else that you intentionally did differently in regards to building fluency from conceptual understanding.
Please note: the prompts below are to help you reflect. There is not an expectation for you to respond to all {or even any} of the provided questions!
Respond:
After reading Chapter 4, please reflect on the questions below and post your response by Monday. Feel free to respond to any of the questions provided or share something else that you intentionally did differently in regards to building fluency from conceptual understanding.
Please note: the prompts below are to help you reflect. There is not an expectation for you to respond to all {or even any} of the provided questions!
Respond:
Developing and Using Equation Strings to Build Toward Fluency
Develop an equation string to use with your students that would support movement toward more advanced use of procedural strategies.
Develop an equation string to use with your students that would support movement toward more advanced use of procedural strategies.
- Select an operation and a specific type of number, such as whole numbers, fractions, or decimals, in which your students need to move toward more advanced use of strategies.
- Identify the mathematics learning goals that will be supported by your equation string.
- Develop a sequence of three to five equations that will build toward student fluency connected to conceptual understanding and reasoning.
- Reflect on the following questions as you plan to facilitate a discussion of the equation string with your students.
- What strategies do you anticipate students using for each equation in your string, including those you hope they will use as examples of more advanced strategies?
- How will you record each strategy using visual models, number bonds, or equations?
- What questions will you ask to probe students' conceptual understanding as you progress though the equation string?
Interact:
On Tuesday, read your colleagues' reflections and respond to at least one other post by sharing a comment, insight, or interesting possibility by Friday.
On Tuesday, read your colleagues' reflections and respond to at least one other post by sharing a comment, insight, or interesting possibility by Friday.
Logging you in...
Susan H · 275 weeks ago
Goals (adapted from page 77, using multiplication vocabulary in place of addition vocabulary):
1. "Students will understand that advanced multiplication strategies use number relationships and the structure of the number system"
2. “Students will understand that numbers can be decomposed and multiplied by parts…”
3. “Students will understand that noticing regularity in repeated calculations leads to shortcuts and general methods for multiplying numbers.”
SEQUENCE:
5 x 10
5 x 20
5 x 30
5 x 32
I can see the majority of the students decomposing the third factor in the equation with the '20.' Some will get to the '30' and feel overwhelmed, but some will understand that they can also decompose this number as well. I do not see many students using the previous product to help them solve the next equation. When they get stuck, I will remind them to use the previous product and hopefully they will subtract it from their current equation, and only add the difference. I am hopeful that some students will realize from the beginning that all of these factors can be decomposed. I would hope they would see 5 x 32 as (5 x 30) + (5 x 2), decomposing the factor 32 into 30 + 2.
RECORDING:
Using an array will help the kids have a very basic concrete visual. Being able to copy and paste the array will hopefully help them see the mathematical basis for the procedures they choose to utilize. I will also use Math Mountains for them to see how the factors can be decomposed. I will also use break-apart drawing lastly, for a more abstract visual when they are moving toward procedural fluency.
QUESTIONS:
Independent Work:
How did you solve for "5 x 10?"
How can you use the product from that multiplication equation to help you solve "5 x 20?"
Can you explain to me why your strategy works?
What is the relationship between 10 and 20?
I see you write 10 x 2 on your paper, where did those numbers come from?
How do you know that 20 is 10 x 2?
If it was a class discussion, I would ask:
Can someone else explain Joe's strategy in their own words?
Can you add on to what Joe said?
Do you agree or disagree with Joe?
Who can use what they know about 5 x 10 to think about the product of 5 x 20?
Who can use what they know about 5 x 10 to think about the product of 5 x 32?
Meribeth Rowe · 275 weeks ago
Meribeth Rowe · 275 weeks ago
Here is the sequence string:
16-9=
26-9=
56-9=
86-8=
I will use the first equation much like a group Number Talk since my students are already familiar with the protocol. I anticipate many students will accurately find the difference and many strategies will be shared. Those strategies may include: counting down 9 from 16, counting up from 9 to 16, a drawing with 16 objects and crossing out 9, or a math drawing representing a ten stick being regrouped into 10 new ones so that 9 ones can be taken away. I don't expect my students to decompose the 9 into a 6 and a 3 and subtract those numbers in two steps, nor do I expect any of my students to think of the 9 as a 10-1 and take away ten then add one back.
I will urge students to use their powers of listening to see if they can find similarities and differences between solutions.
For this string of tasks I will use use a variety of visual representations to help clarify the thinking of specific students leaving them accessible for others to use as they engage in the mathematical thinking of their peers. Open number lines (for counting down, counting up, and decomposition strategies) as well as the familiar ten sticks and ones representational drawings will be used. A laminated hundreds chart will be posted for students that might see patterns as we work through the string of equations much like figures 4.1 and 4.2 in the text.
I am excited to give this a try to help kids move toward greater procedural fluency with other mathematical operations in the future!
On a final note, I wanted to comment on the section titled "Developing Computational Strategies along the Path to Fluency." I really love the definition of fluency as written on page 75 which included student choice, accuracy, and efficiency.
Christine Wilson · 274 weeks ago
Julie Rodriquez · 274 weeks ago
I found a number string today for multiplying decimals that I will send to you through e-mail. :)
I have some questions about giving fluency tests, too. The research mentioned that they can cause math anxiety, which does not surprise me, but does cause me to be concerned about how this affects students overall in their math confidence. I know Renae brought up the topic of fluency tests at our first math leaders meeting, and I would like to revisit this topic again in the future.
Julie
Caty · 274 weeks ago
I totally agree with you about the idea of conceptual then procedural. I think that has been the biggest eye opener for me as a teacher. I was always taught the tricks and until I was teaching math it came full circle with me, and I have always been strong in math growing up. Although sometimes it is challenging to build the conceptual understanding because students do want to jump to the tricks of math but then lack understanding of what is actually happening.
Stephanie Clement · 253 weeks ago
Julie Rodriquez · 274 weeks ago
1. Do not rush conceptual understanding. Although students may demonstrate procedural fluency quickly, if they lack conceptual understanding, it could cause problems in learning mathematics in future classes. I see and hear about this happening too often at the secondary level. Plus the idea of "playing with numbers" opens up so many new avenues for students' understanding with number sense in current and future learning and problem-solving experiences.
2. Fluency embodies many parts - flexibility with solving problems in many ways, ability to understand and communicate how problems are solved, accuracy, and efficiency of solving problems. It is not about memorizing a process or a set of math facts but about the ability to understanding how to solve problems quickly and with clear understanding.
3."Visual representations are an important tool in helping students move to more advanced strategies." Today I had my students solve the sequence of tasks on page 83 and it was eye-opening. Some students embraced the idea of making models of the multiplication problems, while others students complained because they already had an algorithm they wanted to use. I encouraged them to "play with the numbers" and see if they could make sense of the multiplication using a variety of strategies. I had to take a few moments to explain what I had learned from my reading and that I wanted them to slow down and spend time creating representations of their work to make sense of the multiplication. From them on, I learned so much from watching them and listening to their ideas and I feel confident that they learned a lot as well. It was a wonderful experience.
4. I am planning on using a string task for multiplying decimals later this week.
There were many other "stop and jots" I noted as I read, but these were the main ones that stuck with me and that I focused on today as I returned to the classroom.
Susan H · 274 weeks ago
shawnseeleydotcom 44p · 272 weeks ago
The longer I spend in the classroom and tutoring in the summer, the more your first and third points become evident to me. I really see the lack of conceptual understanding and visual representations with students exiting 4th-5th grade. Many students can use an algorithm for a "naked number" problem and arrive at a correct solution, but when asked why/how the algorithm works, they are clueless. I also see this a lot in word problems, when students use the inverse operation and end up with nonsensical solutions like, "93 R3 rows of chairs."
Rachelle Bainton · 272 weeks ago
Jen · 263 weeks ago
Caty · 274 weeks ago
I might try:
8 x 4
8 x 14
8 x 24
8 x 48
180/6
186/6
192/6
174/6
For number strings I do want to encourage mental math, then if they can’t get it then they can make use of their board. I do want them to be able to verbalize what strategies they are using and how they are computing each equation as they move through the string.
shawnseeleydotcom 44p · 272 weeks ago
Several times in this chapter, the importance of visual representations was brought up. I struggle with getting students to create models at times, and I believe it is because they don't really conceptualize what they are doing before they mash numbers together. To address this, I will be implementing a new routine in my classroom, starting this Tuesday, which comes from a book titled "Routines for Reasoning." The routine is called "Connecting Representations." In this routine, I will have students match word problems from the text to visuals (models), and then to equations. One of the word problems won't connect to the models, and one of the equations won't connect to the visuals. Students will work through the routine to make sense of the structure of the comparison word problems (one will be additive and one multiplicative), making connections between chunks of the word problem and chunks of the visual model, then to the equations. Once we've discussed the connections that exist, students will create their own models to represent the remaining, unconnected visual model and equation. I'm believing that this routine, when implemented over time, will help my students deepen their conceptual understanding and help them better progress in their procedural understanding.
shawnseeleydotcom 44p · 272 weeks ago
Rachelle Bainton · 272 weeks ago
Jen · 263 weeks ago
I find myself in a constant battle with taking time to to this great work that lays a child's mathematical foundation and with prepping for tests and meeting deadlines. I also see how critical it is to start this work in kindergarten. If we are able to accomplish this, students would begin their intermediate years with a strong conceptual understanding and number sense. I think we would see more willingness in our students (and teachers) to play with numbers and enjoy math challenges.
Over the years, I have witnessed kids working with word problems and making zero sense of the problem at all. Their goal was simply pulling out numbers to get an answer. I have actual heard kids say, "Two small numbers multiply and one big number and one small number divide." These kinds of statements don't necessarily come from rushing through work or from issues with comprehension, but rather a lack of conceptual understanding. These kids have no visual representation to tie to the mathematical thinking needed to solve the problem. I know I have to do a better job of modeling visual representations and making math models part of our everyday work in order to help kids with their conceptual understanding.
Stephanie Clement · 253 weeks ago