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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Replies to This Discussion

Liz-

Modeling and prompting is so important, especially for  younger grades, we can't give them a task and expect them to know what they are doing. But in the same instance we don't want to model too much for them, it takes away the chance for them to think critically.

Elizabeth 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 , 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. 

1. "Sarah needs to learn to obtain strategies to best support ELL's in math and to feel comfortable and confident in her ability to provide students with the best math education she can."

2.  From what I understand, composite numerosity is when you know that specific numbers represent a quantity and multiplicative reasoning is when you can use your knowledge of composite numerosity to and can add more units to an equation. I know that with multiplicative operations you distribute  one unit over another unit so thats where I got adding more units from. multiplicative is more complex and requires higher level thinking. I'm still learning and have a lot to learn.

In terms of where my thinking is and how I can transfer that with my students, I feel as though I have a better understanding for the differences between additive thinking and multiplicative operations and just knowing that they are different conceptually makes me more aware of how advanced my students thinking is and what kinds of games and activities I can work on with them to get them to multiplicative thinking. 

Mary-

I say do everything electronically. It keeps clutter down and your hand won't get tired!

Mary Shive said:

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.

Cara, 

I also liked that we got a step by step process to transfer the students over to multiplicative thinking. Ron had mentioned that a lot of what we are doing is not about learning things that we can do with our kids tomorrow, but a lot about challenging our thinking, but I really think that by having our ways of thinking of math challenged we have these new ways of developing lesson plans  that better assist our students needs

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,

when it comes to the need statement I can relate to the needing to know how to appropriately utilize a wide variety of resources because I feel like we are getting so much mind blowing information that its hard to know exactly when the right time and in the right ways to use it is, so I can totally see where your coming from in that regard.

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. 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...

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.

Miriah,

Keeping all the data organized is key; I too wish to find a way to do this better this year for my ELL's. I will try to remember to ask Nicola what she does, maybe she has something awesome!

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.

Having formative assessments is important, but you have to prepare them intentionally!

Elizabeth Barnstead said:

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.

Miriah,

Muliplicative reasoning not only helps you prepare effective instructional strategies, but also develop critical thinking skills as Nicola mentioned in the session today.

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.

:)  Electronically!!!

Ashanti Edwards said:

Mary-

I say do everything electronically. It keeps clutter down and your hand won't get tired!

Mary Shive said:

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.

Cara,

As Maggie mentioned, providing MEANINGFUL contexts is what we need to do to help the students move from one dimensional reasoning (additive) to three dimensional reasoning (multiplicative).  

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. 

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