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."

"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.

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 , to the university level. He wants this his to become learning partners and to set learning goals."

The need statement Mary created for me is "Liz needs to examine the amount and type of supports she provides her students, as well as gauge how much modeling/prompting should be done and in what instance."

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 recognizing that a single number represents a group of objects. For example, the number 7 represents a group of 7 ones. Multiplicative reasoning is the ability to start combining composite units into larger units. For example, 4 towers of 5 blocks each represents 20 blocks.  With my students in particular (as a first grade teacher) I think I will need to focus more on being sure that my students have concept of number as well as having a grasp on composite numerocity. 

Amanda,

I also need different ways to assess that are appropriate to what I want my students to show but also practical with the amount of time that I have. I often have to simply use observation with my first graders but I struggle to always get to everybody.
 
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.

1. Miriah needs a way to collect, organize, and appropriately utilize a wide variety if of resources that will support her ELLs in a regular classroom setting.

2. The difference between them is multiplicative reasoning is using more than one composite unit to figure out how many in all, where as the composite unit is like five blocks in a tower and seeing it as five, no longer five individual cubes but as a whole-five. So the difference in developing these are you need to understand what composite units are before you can move on to multiplication.

1. Share the needs statement that your Thinking Partner developed for you.

The needs statement Miriah created for me says, "Amy needs a way to meet her students needs individually and not just group them together as "one level" but utilize their strengths to meet each individuals' needs.

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 me, at least my understanding thus far, of developing composite numerosity involved the ability to group individual units into another unit and have an awareness of that new unit without having to go back and count the separate units of one that make it up. For example, putting together 5 cubes; there are 5 separate units of one put together. Composite numerosity would entail counting all 5 and then understanding that it is not 5 and not 1...2...3...4...5.

Multiplicative reasoning would be taking a step up and being able to group the composite units into a different composite unit. For example, 5 towers of 5 cubes is a new composite unit of 25. Not having to go back and count 25 separate units would be part of this.

Miriah,

I think all of us could use your "need statement". I am working on building my "tool box" of strategies that will help support my ELL's in the classroom.
 
Miriah Bruns said:

1. Miriah needs a way to collect, organize, and appropriately utilize a wide variety if of resources that will support her ELLs in a regular classroom setting.

2. The difference between them is multiplicative reasoning is using more than one composite unit to figure out how many in all, where as the composite unit is like five blocks in a tower and seeing it as five, no longer five individual cubes but as a whole-five. So the difference in developing these are you need to understand what composite units are before you can move on to multiplication.

1.  Cara needs a way to sustain and bring over what she is hearing and discussing during the workshop into her upcoming classroom experiences.

2.  How I understand the difference (and how it has developed) between developing "composite numerosity" and "multiplicative reasoning":  I am still trying to wrap my brain around this new way of thinking but I think it is finally starting to set in.  Composite numerosity is the understanding that single units can be combined into a new "single" unit made of a number of parts.  Multiplicative reasoning takes these composite units and compiles them into a new unit. 

What I need to focus on with my students:  Since multiplicative reasoning is such a big conceptual leap from additive reasoning I must assist my students' development of the new concept.  I want to allow them to discover the concept on their own through the use of games such as PBM and purposeful questions into their reasoning.  I especially appreciated the progression that we can use as teachers; we can start with the game, then play the game at tables while hiding the towers with paper, then guiding the game with word problems, then taking the "game" away and allowing kids to imagine the game with pictures, then simply having students answer the math questions by just reading the words.  It is a step-by-step scaffold process in which we can bridge their previous knowledge of a single composite unit to the new knowledge we want them to develop of a compilation of composite units (multiplicative reasoning). 

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. 


Cara,

I think you make a great point about allowing students to develop the concept on their own. If we use terms like multiplicative reasoning or composite numerocity without providing background knowledge (through experience) the words mean nothing. We need to create a context where the words make sense and is meaningful. Creating classroom experiences, like the game creates a context for learning that everyone in the class has experienced, and understanding can be built from that. Great point!

Maggie 
Cara Russell said:

1.  Cara needs a way to sustain and bring over what she is hearing and discussing during the workshop into her upcoming classroom experiences.

2.  How I understand the difference (and how it has developed) between developing "composite numerosity" and "multiplicative reasoning":  I am still trying to wrap my brain around this new way of thinking but I think it is finally starting to set in.  Composite numerosity is the understanding that single units can be combined into a new "single" unit made of a number of parts.  Multiplicative reasoning takes these composite units and compiles them into a new unit. 

What I need to focus on with my students:  Since multiplicative reasoning is such a big conceptual leap from additive reasoning I must assist my students' development of the new concept.  I want to allow them to discover the concept on their own through the use of games such as PBM and purposeful questions into their reasoning.  I especially appreciated the progression that we can use as teachers; we can start with the game, then play the game at tables while hiding the towers with paper, then guiding the game with word problems, then taking the "game" away and allowing kids to imagine the game with pictures, then simply having students answer the math questions by just reading the words.  It is a step-by-step scaffold process in which we can bridge their previous knowledge of a single composite unit to the new knowledge we want them to develop of a compilation of composite units (multiplicative reasoning). 


Miriah, 

Your needs statement really resonated with me. I often say, I have so many resources but don't know how to group them. Sometimes I feel like my resources and ideas get lost in a pile of paper, and I just can't locate them easily when I want them. That being said, I think tools such as the binder that was created for this institute are a good idea. Perhaps by creating binders that are specific in the content they contain will help you keep your ideas and thoughts organized and easily accessible. 

Maggie


Miriah Bruns said:

1. Miriah needs a way to collect, organize, and appropriately utilize a wide variety if of resources that will support her ELLs in a regular classroom setting.

2. The difference between them is multiplicative reasoning is using more than one composite unit to figure out how many in all, where as the composite unit is like five blocks in a tower and seeing it as five, no longer five individual cubes but as a whole-five. So the difference in developing these are you need to understand what composite units are before you can move on to multiplication.

1. "Mary needs a way to structure keeping a log of notes about her students.  (What will it look like?  What is the purpose?  How will it be organized?)"

2. My understanding of composite numerosity and multiplicative reasoning is that composite numerosity is the first step toward multiplicative reasoning.  What I mean by this is that in multiplicative reasoning, you are taking several composite units to create a new composite unit, or in other words counting the composite units knowing what each unit represents.  This requires students to shift from counting ones to counting composite units.  I believe Ron also referred to this process as multiplicative double-counting (ex. one finger=3 units, 2 fingers=6 units, and so forth).  However, it is important that the students recognizes where they should stop.

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."

Ashanti needs a way to create the assessments before she enters the classroom to generate great ideas before hand.

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 numbers are numbers that have that smaller factors that multiply to equal them. Composite numerosity is the study of how these numbers are made of these smaller parts. Multiplicative reasoning is understanding that a certain number of groups all containing the same amount equal a certain number.  This certain number that these parts equal is a composite number.  Obviously then if you take the factors of the composite number, then you can use multiplicative reasoning to verify that the composite number was factored correctly. Because of this, the students should first learn multiplicative reasoning. The students can understand multiplicative reasoning without knowing what a composite number is. However, in order to understand, or at least find a composite number, they need to first know how to multiply.  For example, "is 30 a composite number?" It can be broken down into 2,3, and 5.  For me to do that in my had I thought 30 can be broken down into 5 and 6. I knew this because I had first learned to multiply and I knew 5 times 6 is 30.  I did the same thing with 6 to get the 2 and the 3. So with my students , I need to focus on multiplicative reasoning first because that is the base.

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