Today was our 3rd day. We began the day with Shaw reading the EOD, followed by you working with your ThinkingPartner in the explore phase to identify a need you have, that you want to try out in the fall regarding math and SLA.

Ron's 1st session involved developing the need for multiplicative reasoning by distinguishing between additive and multiplicative, with multiplicative having the need for the equal distribution of units across a composite unit. This was illustrated with the green and yellow unifix cubes representing limes and lemons.

Ron's 2nd session involved engaging in Please Go Bring Me in order to illustrate how to help students develop multiplicative reasoning by:

1. Running to get the cubes, pushed learners to develop a composite number, and the repeated "Please Go Bring Me" forced the bringer to have a place holder in her mind that "six units" will always be the same height as the previous six. So, that composite numerocity was developed while fetching the cubes. And the fetching took place in a different part of the room.

2. Then the need for multiplicative reasoning took place when the "sender" asked the bringer to total the entire amount of towers.

Please do two things:

1. Share the needs statement that your ThinkingPartner developed for you. For example, Amanda Rossini created for me: "Ralph needs a way to bridge previous ways of using formative assessments he used when teaching 3rd grade, to the university level. He wants this his university students to become learning partners and to set learning goals."

2. Explore how you understand the difference (and how it is developed) between developing "composite numerocity" and "multiplicative reasoning"...and where your thinking is about what you need to focus on with your students.

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1. Share the needs statement that your ThinkingPartner developed for you. For example, Amanda Rossini created for me: "Ralph needs a way to bridge previous ways of using formative assessments he used when teaching 3rd grade, to the university level. He wants this his university students to become learning partners and to set learning goals."

The statement from Cecilia was: "Adam needs a way to use the developmental math stages as a 'map' to help guide him to teaching with a purpose.  This will help him set developmentally appropriate goals, starting points and thoughtful activities for individual students in order to minimize 'wasting time'." 

2. Explore how you understand the difference (and how it is developed) between developing "composite numerocity" and "multiplicative reasoning"...and where your thinking is about what you need to focus on with your students.

As far as I understand, Composite numerocity is the ability for one to identify a number represented by figurative counters and having a firm grasp of a one to one correspondence, then using these understandings to represent a larger number (for example, 5 blocks equals one stack of 5).  If there is an understanding of composites, this number is assumed and does not need to be recounted.  When considering Multiplicative Reasoning, it is a prerequisite for students to use composite numerocity to make multiple groups (for example, multiple stacks of 5) and then determining the number that the composite groups represent.  

Teaching with purpose will make difference!!

Adam Barnstead said:

1. Share the needs statement that your ThinkingPartner developed for you. For example, Amanda Rossini created for me: "Ralph needs a way to bridge previous ways of using formative assessments he used when teaching 3rd grade, to the university level. He wants this his university students to become learning partners and to set learning goals."

The statement from Cecilia was: "Adam needs a way to use the developmental math stages as a 'map' to help guide him to teaching with a purpose.  This will help him set developmentally appropriate goals, starting points and thoughtful activities for individual students in order to minimize 'wasting time'." 

2. Explore how you understand the difference (and how it is developed) between developing "composite numerocity" and "multiplicative reasoning"...and where your thinking is about what you need to focus on with your students.

As far as I understand, Composite numerocity is the ability for one to identify a number represented by figurative counters and having a firm grasp of a one to one correspondence, then using these understandings to represent a larger number (for example, 5 blocks equals one stack of 5).  If there is an understanding of composites, this number is assumed and does not need to be recounted.  When considering Multiplicative Reasoning, it is a prerequisite for students to use composite numerocity to make multiple groups (for example, multiple stacks of 5) and then determining the number that the composite groups represent.  

1.  Joe wrote,  "Lisa needs to develop effective ways to differentiate instruction for her students that combines what the CCSS require and what her students actually need according to their ACCESS scores and background knowledge.

2.  My understanding of composite numerocity is the knowledge that a single number represents a group of objects no matter where that number appears or if the objects are not even present.  For example,  7 is 7 is 7.  That doesn't change, even though the mathematical situation may.  The student does not need to go back and recount the objects to arrive at 7.  They have the "concept of number" to know what 7 represents. 

Multiplicative reasoning can only occur when a child has developed composite numerocity.  In multiplicative reasoning one unit is distributed over another unit to produce a third unit that is different.  For example if you had 3 oranges in each bowl and you had 4 bowls, the total number of oranges would be the new unit or 12 oranges.   We developed our understanding of this today when we played PGBM with the cubes and towers.  

What hit me over and over again today was the importance of observing our students and asking them to explain their thinking to us.  I know in my past experiences, I have not done enough of this.  As I prepare to work with new students this year, this will be forefront in my mind.  I keep hearing Ron's words, "Math is reasoning."  My job is to provide a learning community where my students are willing to share what their thinking strategies are.  With my ELL students, it is especially important for me to support them with the vocabulary they need to do this.  

Amanda,

  In terms of your number 2 response, this was something that we discussed at our table.  We are now aware of several activities that "do not effectively teach" multiplicative reasoning or pre-numerical/numerical reasoning, but only two activities that do.  I wonder if this may be brought up tomorrow to get Ron's ideas of what else may be done that will be effective?

Amanda Rossini said:

1. Share the needs statement that your ThinkingPartner developed for you. For example, Amanda Rossini created for me: "Ralph needs a way to bridge previous ways of using formative assessments he used when teaching 3rd grade, to the university level. He wants this his university students to become learning partners and to set learning goals."

"Amanda needs a way to learn and adminster, in a systematic way, formative assessments that are multi-modal in nature and a way to document and collect."

2. Explore how you understand the difference (and how it is developed) between developing "composite numerocity" and "multiplicative reasoning"...and where your thinking is about what you need to focus on with your students.

The composite numeral  is where a object stands for a particular number and the unit stays the same. In multiplicative reasoning there are two units that are used in a distribution way. I am thinking about using this game whole group and then reteaching it in small intervention groups. I am wondering what other activities that I can use and/or create for this.

Lisa,

  I think it's great that you brought up using the Access scores to help guide, plan and differentiate instruction.  I think this is a great goal and something that I plan on working on in the upcoming weeks as well.

Lisa Hogbin said:

1.  Joe wrote,  "Lisa needs to develop effective ways to differentiate instruction for her students that combines what the CCSS require and what her students actually need according to their ACCESS scores and background knowledge.

2.  My understanding of composite numerocity is the knowledge that a single number represents a group of objects no matter where that number appears or if the objects are not even present.  For example,  7 is 7 is 7.  That doesn't change, even though the mathematical situation may.  The student does not need to go back and recount the objects to arrive at 7.  They have the "concept of number" to know what 7 represents. 

Multiplicative reasoning can only occur when a child has developed composite numerocity.  In multiplicative reasoning one unit is distributed over another unit to produce a third unit that is different.  For example if you had 3 oranges in each bowl and you had 4 bowls, the total number of oranges would be the new unit or 12 oranges.   We developed our understanding of this today when we played PGBM with the cubes and towers.  

What hit me over and over again today was the importance of observing our students and asking them to explain their thinking to us.  I know in my past experiences, I have not done enough of this.  As I prepare to work with new students this year, this will be forefront in my mind.  I keep hearing Ron's words, "Math is reasoning."  My job is to provide a learning community where my students are willing to share what their thinking strategies are.  With my ELL students, it is especially important for me to support them with the vocabulary they need to do this.  

Mitzy,

Your needs statement about communication with the ELL teacher really struck me.  As I prepare to work with Marvin's K, 1 and 2 ELL students, I realize how important it is that I communicate all that I know about my students with their classroom teachers.  I want the W-APT and ACCESS scores to be meaningful for them so that they can use this information to guide their work with their students.  Collaboration is essential between the classroom and ELL teacher.
 
Mitzy Barnstead said:

1. Share the needs statement that your ThinkingPartner developed for you.

 

Sarah wrote for me, "Mitzy needs a way to find time to communicate with the ELL teacher to understand scores & needs of her students so she can adjust her lesson plans to better suit their needs. Also, a way to organize the new found information.

2. Explore how you understand the difference (and how it is developed) between developing "composite numerocity" and "multiplicative reasoning"...and where your thinking is about what you need to focus on with your students.

Composite numerocity is when students recognize that a number represents a group of items; such as 5 representing five  unifix cubes. This group of 5 is now a "unit" of cubes. The student can look at the unit of cubes and know it is five, and if you were to cover it with paper, they would still say it is 5 without having to count each cube.

Multiplicative reasoning is when a student can look at the unit of 5 cubes and see there are 6 "towers" or groups of 5 cubes, and be able to identify that as a new unit of 30 total cubes. The students should not have to count each individual cube, they should be able to think, "there are 6 towers of 5, so 5, 10, 15, 20, 25, 30...".

My focus with my students will be on questioning them appropriately and requiring them to always explain their thinking either verbally or written (or both). We actually started this across the subject areas last year due to the shift to Common Core in our building. By hearing their thinking, I can identify if they actually have the mathematical reasoning, or used computation without really knowing why...

:))

Lisa Hogbin said:

Mitzy,

Your needs statement about communication with the ELL teacher really struck me.  As I prepare to work with Marvin's K, 1 and 2 ELL students, I realize how important it is that I communicate all that I know about my students with their classroom teachers.  I want the W-APT and ACCESS scores to be meaningful for them so that they can use this information to guide their work with their students.  Collaboration is essential between the classroom and ELL teacher.
 
Mitzy Barnstead said:

1. Share the needs statement that your ThinkingPartner developed for you.

 

Sarah wrote for me, "Mitzy needs a way to find time to communicate with the ELL teacher to understand scores & needs of her students so she can adjust her lesson plans to better suit their needs. Also, a way to organize the new found information.

2. Explore how you understand the difference (and how it is developed) between developing "composite numerocity" and "multiplicative reasoning"...and where your thinking is about what you need to focus on with your students.

Composite numerocity is when students recognize that a number represents a group of items; such as 5 representing five  unifix cubes. This group of 5 is now a "unit" of cubes. The student can look at the unit of cubes and know it is five, and if you were to cover it with paper, they would still say it is 5 without having to count each cube.

Multiplicative reasoning is when a student can look at the unit of 5 cubes and see there are 6 "towers" or groups of 5 cubes, and be able to identify that as a new unit of 30 total cubes. The students should not have to count each individual cube, they should be able to think, "there are 6 towers of 5, so 5, 10, 15, 20, 25, 30...".

My focus with my students will be on questioning them appropriately and requiring them to always explain their thinking either verbally or written (or both). We actually started this across the subject areas last year due to the shift to Common Core in our building. By hearing their thinking, I can identify if they actually have the mathematical reasoning, or used computation without really knowing why...

Maggie,

I love what you said at the end of your second response!  "Through discourse, they may develop new ways of looking at problems, and work out issues they may have. This strategy will create a sense of community, create a collaborative learning environment, and give me insight as to the way my students are absorbing concepts I introduce."  You hit all the essential components of a classroom where students are given the opportunity to take ownership of their thinking and  feel safe to take risks and share their thinking with others.  With your insight and observations as their teacher, there is going to be some incredible learning taking place in your classroom!  
 
Margaret Fairless said:

1. Share the needs statement that your ThinkingPartner developed for you. For example, Amanda Rossini created for me: "Ralph needs a way to bridge previous ways of using formative assessments he used when teaching 3rd grade, to the university level. He wants this his university students to become learning partners and to set learning goals."

Donna created the following needs statement for me: "Maggie needs a way to teach math as a third language while incorporating second language acquisition skills; also, motivating students intrinsically to do math." 

2. Explore how you understand the difference (and how it is developed) between developing "composite numerocity" and "multiplicative reasoning"...and where your thinking is about what you need to focus on with your students.

To be honest, I'm still in the process of wrapping my mind around these two ideas. From my understanding, composite numerocity is the idea that the number 7, for example, stands for a group of seven individual 1's. A person demonstrates the concept of  composite numerocity, when they do not need to go back and recount numbers they've already counted. They don't need to recount because they understand that the single established number (the composite) represents the individual parts they previously counted. Multiplicative reasoning is when someone has the ability to combine individual composite units to create another composite unit. For example, in the game presented in class, students were asked to bring 5 towers, composed of 4 cubes each. 5 and 4 are both composite numbers. Students have multiplicative numerocity when they take these composite units and compose a new composite unit. 

The biggest thing I took away from today is the fact that in order to assess learning, teachers need to focus on a students mathematical reasoning more so than the answer. Students can get to the wrong answer and still know the process and have sound reasoning. On the other hand, students can get the correct answer and have no understanding of how they arrived at the answer. Correct reasoning, not the correct answer, indicates true understanding of the concepts. In addition to changing my focus from having students arrive at the correct answer to having the students understand the process, I need to allow students to engage in discourse with each other about their reasoning. Through discourse, they may develop new ways of looking at problems, and work out issues they may have. This strategy will create a sense of community, create a collaborative learning environment, and give me insight as to the way my students are absorbing concepts I introduce. 

1.

Ellis needs a way to practically and realistically implement these strategies into a real classroom as a new teacher.

2.

I'm going to admit that I don't have any concept of either. Sorry. I can't really tell you what's the difference. And I'm tempted to go down and read other's comments but then I'd be plagiarizing their ideas. The second part of this problem asks what I need to focus on with my students and the simplest answer would be to say that I need to get the concepts. I think I have time to develop these concepts. I'd like to think that this weeks workshop is planting seeds and my ideas and thoughts need more time (and probably a lot more coaching) before they produce useable fruit.

Mitzy, I want to "Yes-And..." build on your thinking: In addition to building "reasoning" into your math instruction, daily...what would happen if in other content areas you asked questions like "how is what we're figuring out in science, like what were figured out in math? How is what we're are working through while we read this tex, or write this piece, like what we know how to do in math?"

Explicitly asking kids to transfer how they approach reasoning in one setting to a different one, pushes them, as Vygotsky tells us to re-learn, unlearn, build upon what they "formulated" in one setting, and then "reformulate" it in a different setting. And that's when powerful growth occurs.


Mitzy Barnstead said:

Amanda,

I too would like to use the game to aid in mathematical reasoning. I would like to institute a "Reasoning Day" in math where for one day a week we focus on one problem which requires students to think, discuss, and reason to find an answer to a problem. I think it would switch things up and the one full day intro to the game would help teach it to everyone so it can be used later for small groups.

Amanda Rossini said:

1. Share the needs statement that your ThinkingPartner developed for you. For example, Amanda Rossini created for me: "Ralph needs a way to bridge previous ways of using formative assessments he used when teaching 3rd grade, to the university level. He wants this his university students to become learning partners and to set learning goals."

"Amanda needs a way to learn and adminster, in a systematic way, formative assessments that are multi-modal in nature and a way to document and collect."

2. Explore how you understand the difference (and how it is developed) between developing "composite numerocity" and "multiplicative reasoning"...and where your thinking is about what you need to focus on with your students.

The composite numeral  is where a object stands for a particular number and the unit stays the same. In multiplicative reasoning there are two units that are used in a distribution way. I am thinking about using this game whole group and then reteaching it in small intervention groups. I am wondering what other activities that I can use and/or create for this.

Please same more about "practically" and into a "real" classroom?

Ellis Anderson said:

1.

Ellis needs a way to practically and realistically implement these strategies into a real classroom as a new teacher.

2.

I'm going to admit that I don't have any concept of either. Sorry. I can't really tell you what's the difference. And I'm tempted to go down and read other's comments but then I'd be plagiarizing their ideas. The second part of this problem asks what I need to focus on with my students and the simplest answer would be to say that I need to get the concepts. I think I have time to develop these concepts. I'd like to think that this weeks workshop is planting seeds and my ideas and thoughts need more time (and probably a lot more coaching) before they produce useable fruit.

1.  Lisa Hogbin wrote this need statement for me.  

  • "Joe needs to develop questioning techniques to probe his students thinking"

2. From my understanding, developing composite numerocity and multiplicative reasoning starts with activities that don't prohibit students' background knowledge, but encourages their background knowledge.  It starts with activities that allow students to take ownership of their mathematics and expects justification of solutions.  This is where I feel I need to focus on with my future students.  I want to start with where they are and incorporate activities like PGBM that will give insight into the processes that students go through to get to their answer.  

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