UBD Template: Organizing and Collecting: The Number System

Stage 1 – Desired Results

Established Goals:

Count to 120, starting at any number less than 120. In this range, read and write numerals and

represent a number of objects with a written numeral. (1.NBT.A.1)

Understand that the two digits of a two-digit number represent amounts of tens and ones.

Understand the following as special cases: (1.NBT.B.2)

10 can be thought of as a bundle of ten ones — called a “ten.” (1.NBT.B.2a)

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven,

eight, or nine ones. (1.NBT.B.2b)

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven,

eight, or nine tens (and 0 ones). (1.NBT.B.2c)

Understandings:

Students will understand that …

Students will understand that compensation

and equivalence exist (i.e. 5 + 3 = 4 + 4) (Liu,

Dolk, & Fosnot, 2007).

Students will understand the concept of

unitizing (i.e. being able to see three groups of

five within 15 objects) (Liu, et. al., 2007).

Students will understand that place determine

value, not digit (i.e. 5 can represent 5 ones, 5

tens, 5 hundreds, etc.) (Liu et. al., 2007).

Students will understand that numbers can be

composed and decomposed in a variety of

ways, which will begin to lay a foundation for

understanding commutativity and associativity

(Liu et. al., 2007).

Essential Questions:

How do we count a group of objects?

Are there different ways to count objects?

Can we group items differently within one

large total?

How many ways can we make ten?

Are there multiple ways to create a certain

number of objects?

What is place value?

Can we see patterns among numbers when

we count them?

Demystifying Mathematics

This purpose of this unit is to develop number sense, counting, and the concepts of

place value. This unit strives to push students to think about numbers in a new way. Before

entering this unit, most students will be counting by ones. The activities in the unit push the

students to group numbers in new ways, to examine patterns within their counting, and to

begin to think about addition and subtraction using friendly numbers (5 and 10). It attempts to

create an authentic learning situation by having students take an inventory of their own

classroom supplies and tries to engage them by having them be responsible for a specific item

and giving them plausible ideas about the need to order new materials based on their

inventory. In order to fully develop in this area of math, students should be able to count by

ones, be able to work in partner or small group settings, be able to represent their counting by

writing digits, and be willing to try out new strategies.

Data-Driven Instruction

Prior to starting this unit, we did administer some preliminary assessments. In one

assessment, we asked students to write their numbers as high as they could in counting order.

In the other assessment, we tested the students’ ability to hear a number told to them and

record it correctly. While we did both of these assessments before starting our math unit, we

never did anything with the data. I am not sure if this is because my lead teacher did not think

it would be helpful or if she was unsure of what to do with the data. While I do not have any

real assessment of the data, I can give some of my observations. Most of my students, around

80%, could write their numbers up to 50. Many of them have problems with numeral reversal,

which is common in first grade. Of the 20% who could write their numbers up to 100, they all

struggled after that. For example, instead of writing “101”, they would write “1001”. This

shows that they have no concept of place value when they get into the hundreds. One

interesting observation from the first assessment was that even if students could not write their

numbers up very high, they would all write “100” at the bottom right hand corner, which was

the last box on the first page. My lead teacher and I attributed this to the students being

exposed to the hundreds chart and seeing the “100” in the bottom right corner of the chart

every day in calendar math. In the second assessment, students had a hard time writing the

numbers correctly, mostly due to the number reversal. For example, if my lead said the

number “83”, some students would record the number as “38”.

We used three summative assessments for this unit. During the summer, the first grade

team created benchmark assessments based on the first grade math standards with the help of

our school’s math consultant. One assessment asked the students to explain what the “2” and

the “8” represented in the number “28” using pictures or words. The second assessment was a

story problem. It asked the question, “Stephanie had 60 crayons. They fit into boxes of 10.

How many boxes did she have all together?”. The third assessment asked students to compare

two numbers that use the same digits (i.e. 45, 54) and figure out which number is bigger and

why. The assessments can be found in Appendices A, B, and C respectively.

I am unsure of why we did not do more pre-assessment or why we did not analyze the

data that we did have. Looking back, I think it would have been helpful to give each student a

prescribed number of objects and ask them to count them. I think having anecdotal

information on how they count beforehand could have been helpful. It also might have been

interesting to ask them some preliminary information about place value before beginning the

unit or ask them to skip count by tens or fives. These ideas are based solely on my classroom

experience, as I have not been given many resources on preliminary assessments to use.

Lesson Narratives

Day Day One Day Two Day Three Day Four Day Five Day Six Day

Seven

Objective SWBAT count a quantity of classroom supplies and record their counting on a piece of

SWBAT count a group of objects and will begin to grasp the idea of grouping items into friendly number

SWBAT group their supplies into packs of ten, record the number of ten-packs and loose ones, and have a basic understandi

SWBAT notice patterns within numbers related to place value.

SWBAT discuss the pattern in three digit numbers and they will explore the parts of ten.

SWBAT learn and develop fluency with the parts of ten when using two or more addends.

SWBAT make ten as an addition strategy and unitize a group of ten.

paper. groups. ng of a ten-frame.

Activity After reading The Masloppy Family, students will begin to investigate counting and place value by taking an inventory of the materials in the classroom.

Students continue to count their materials and are hopefully using strategies implemented by their peers. They are hopefully moving towards grouping objects.

Students are introduced to a ten-frame and the idea of grouping objects into packs/bundles of ten. Students are urged to begin to bundle/pack their materials into groups of ten. Students record number of ten-packs and number of loose ones for their inventory materials.

The class creates an anchor chart together, documenting the class’ inventory, with a breakdown of total, tens, and ones. Students look for patterns within the chart.

Students add the inventory totals of the math manipulatives to the chart, which allows them to examine patterns within three-digit numbers. Students also think about parts of ten as they decide how many more loose ones would need to be ordered to complete a pack of ten for their inventory material.

Students examine patterns on their inventory chart when looking at their numbers of added inventory from the previous day. Students also play the Rolling for Tens game, where they learn and practice parts of ten using at least two addends.

Students play Collecting Stamps, which focuses on the skill of thinking of ten as ten items and as one group of ten.

Assessme

nt

Observe students’ counting and grouping strategies. Take dated observation notes for

Continue to observe and record how students are counting and grouping objects.

Counting skills can be assessed, especially skip counting by tens and unitizing objects. Document these skills

Not applicable

Anecdotal notes on students’ knowledge on parts of ten would be a helpful tool to have moving forward.

Observe the combinations students use when playing Rolling for Tens. Observation notes for the

Observe students as they play the game. Note any changes in strategies that

students’ portfolios.

in anecdotal notes.

portfolio will be helpful.

occur. Having extra copies of the game board will allow you to shade in the landmarks made by each child and can be helpful evidence for the portfolio.

Materials

Please see Appendices A, B, and C. for copies of the place value benchmark assessments

given at the end of this unit. See Appendix D for materials needed for the lessons. Other

materials are listed in each day’s lesson plan.

References

Liu, N., Dolk, M., & Fosnot, C. (2007). Organizing and collecting: The number system.

Orlando, FL: Firsthand Heinemann.

Analysis of Student Learning

Unfortunately, the rubrics for our assessments have not been created yet. This will

most likely happen in early January. Therefore, I cannot give any solid data about my students’

mastery of place value. I can, however, give some observations made when l looked over the

assessment data. Also, at the time of this analysis, only one of the three assessments had been

given to the students. The place value assessment in Appendix A was given on December 6 and

7. The place value assessments in Appendices B and C will be given within the next week and a

half.

As I looked over the completed place value assessments, I noticed lots of interesting

things about my students’ understanding of place value. I estimate that about 80% of my class

demonstrated an understanding of place value through pictures, words, or both. For example,

many students drew pictures of place value tools (sticks of ten, cubes of one) or unifix cubes

(stacks of ten, individual cubes) to show what the 2 and the 8 represented in the number 28.

Some students also drew boxes of 10 to represent the 2 tens and used individual crayons to

represent the 8 ones. When using words to explain their drawings, many students used the

wording of “tens place” and “ones place”, as well as “2 tens” and “8 ones”. Typically, the

aforementioned pictures and phrases were used together.

Some students demonstrated an understanding of place value through their drawing or

their words, but were unable to show their understanding through both modes of

communication. For example, one student drew two sticks to represent the two tens and drew

eight circles to represent the ones place, which is a correct drawing. However, in her sentence,

the student wrote, “The 2 represents sticks and the 8 represents apples”, which shows that she

does not fully understand what her pictures are representing in relation to math. We also had

students who could articulate through their sentence what they meant but did not accurately

represent their thinking in a drawing. One student drew two crayons to represent the two tens

and eight crayons to represent the eight ones, but was able to write a sentence that said that

there were two tens and eight ones, which was not shown in the picture.

We also had some students who were close to grasping the concept of place value, but

just were not there yet. For example, one student drew two boxes of ten to represent the two

tens and drew eight boxes of ten to represent the eight ones. The student knows that she

should use tens to represent something, but does not quite understand what her drawing

means. We also had a few students who reversed the place value spots. On one assessment,

the student drew two ones and eight tens. She drew the correct pictures, but they were on the

wrong side of the paper. She mixed up where the tens place and the ones place were.

I wanted to share some thoughts about the assessment in relation to the unit. My first

grade team had a hard time getting started with math this year. We had an unclear math map

to use and it was very disorganized and was not laid out well. My lead teacher and I began to

teach math that we thought was relevant and important for our students, but we were not

following a thought out sequence and our classroom during math looked very different from

the other first grade classroom. Eventually, the math map was revised, and while my lead

teacher and I were making our own choices about math and moving in what we felt was a

thoughtful and constructivist direction, we were mandated to use this curriculum in our

classroom. Because of that, there was not much thought put into using this unit and how it

aligned with the benchmark assessments. It seemed that we were using the curriculum

because it was an available resource. Because of that, the assessments do not give a

completely accurate view of how the unit works as a whole.

Objective Analysis of Teaching Episode

Positive ClimateRelationships, positive affect, positive communication, respect

I would give myself a 5 on the CLASS in Positive Climate. I often showed my students respect while they were speaking by using active listening behaviors such as eye contact, nodding my head as they spoke, and repeating what they said back to them. At times, my students were respectful of one another while one student was talking (i.e. listening to that student speak without talking), but there were instances where side conversations continued when I would call on another student and some students were also playing around on the carpet. Lastly, I thought that I gave positive feedback to students about their thinking. I would describe what they did after they answered and would add some positive feedback after that.

Negative ClimateNegative affect, punitive control, sarcasm/disrespect, severe negativity

I gave myself a 2 in Negative Climate. I did show some points of irritation when I was teaching (i.e. when other students would talk to their neighbors after I had called on a student to share their thinking with the group), but I did not think that my irritation lasted very long. I felt that I used calm ways of handling the misbehavior. I did not yell at my students. I tended to use the phrases “I’ll wait”, “Show me you’re ready”, and just giving wait time when the students were misbehaving. I did call out one student, Natalie, and asked her what a fellow student had just shared with the group. I did this because I had noticed that she was talking to her neighbor consistently throughout part of the lesson, but I realize that it might not have been the best way to respond to that.

Teacher SensitivityAwareness, responsiveness, addresses problems, student comfort

I gave myself a 4 for Teacher Sensitivity. I think that my students are kind of locked into roles of whether or not they raise their hands and share. The same students raise their hands and the same students do not raise their hands. I am not always fair about who I call on for students. I tend to anticipate in my mind about who will have the right answer or who needs to improve on this skill and I call on students appropriately. I think that I sometimes support students who give wrong answers by trying to question them in a way to nudge them towards the correct answer, but sometimes I just move on to another student without having a student explore their thinking more deeply.

Regard for Student Perspectives Flexibility and student focus, support for autonomy and leadership, student expression, restriction of movement

I gave myself a 3 for Regard for Student Perspectives. My classroom is fairly teacher controlled. Students have assigned desk spots and assigned carpet spots. They are also expected to sit a certain way on the carpet. We call it “student position”, which means students are sitting up straight, with their legs crisscross, voices off, and hands folded in their laps. There was a fair amount of teacher talk versus student talk during my lesson. I felt that I was leading the lesson, but I was asking the students a lot of questions and engaging their thinking. This was also a whole group instruction time with students primarily responding to teacher questions, to they were not really involved with the conversation or lesson as much as they could have been.

Behavior ManagementClear behavior management, proactive, redirection of misbehavior, student behavior

I gave myself a 6 in Behavior Management. I think that I consistently gave clear expectations, both for behavior and for the activity. I would not start the activity or I would wait if the students were not showing me they were ready to move forward. I would remind students of the expectations of the activity frequently. I typically used subtle behavior clues (i.e. “teacher” look, gentle touch, eye contact, head shake) to show students that they needed to fix what they were doing. I was not always aware of all of my students. I noticed that some of my students were lying around or messing around with friends during the lesson and some of those issues were never addressed.

ProductivityMaximizing learning time, routines, transitions, preparation

I gave myself a 4 in Productivity. I felt that I was mostly prepared for my lesson. I had my ten-frames handy, my book was nearby, and I had the materials that some of my students modeled with near me. However, I could have had my chart flipped to the correct page and had the page in the book marked ahead of time. I felt that I had moderate pacing. My mini-lesson ran about 17 minutes long. I could tell from the movement on the carpet near the end that I was taking a little too long with everything and that they were ready to move on. I think that my students show a strong knowledge of our carpet routine. They know how to act and how to sit. They know what is expected of them, and for the most part, they follow that routine.

Instructional Learning Formats Effective facilitation, variety of modalities and materials, student interest, clarity of learning objectives

I gave myself a 3 on Instructional Learning Formats. It was a heavily auditory lesson with lots of question/answer and discussion involved. Some visuals were used, but no other modalities were really utilized. I thought that there was moderate student engagement. I kept their attention for most of the lesson, but began to lose them as they sat on the carpet for too long. The learning objective was somewhat clear, but I’m not sure my students could have told you what the point of the lesson was.

Concept DevelopmentAnalysis and reasoning, creating, integration, connections to real world

I gave myself a 4 in Concept Development. I thought that I asked many how and why questions. After students gave an answer, my typical follow-up was “why do you think that?” or “what makes you think that?” There were some real world connections for the students through the idea of inventorying the classroom to keep track of our materials in case they get lost or taken, but I am not sure how authentic that connection was for them. I did not think that there was much creativity or generation happening from the students. I was asking questions that typically required one answer. I did ask them to explore and explain their thinking, but I do not think we were doing anything very creative or student-generated.

Quality of FeedbackScaffolding, feedback loops, prompting thought processes, providing information, encouragement and affirmation

I gave myself a 6 in quality of feedback. I felt that I did a pretty good job of providing hints or asking nudging questions to students when they would respond with an answer. I tried to push them to examine their thinking more. I almost always asked the students to explain their thinking to the class. They could not just answer the question correctly, but they had to explain how they figured it out. I also think I did a good job of repeating and then expanding on student answers so that the class could learn from their peer.

Language ModelingFrequent conversation, open-ended questions, repetition and extension, self- and parallel talk, advanced language

I gave myself a 5 in Language Modeling. I thought that I used a good mix of closed and open-ended questions. There was typically a “right” answer to the questions, but I tried to use open-ended questions so students could be metacognitive. I would often repeat and extend student responses so that we could all learn and benefit from one person’s answer. I used some advanced language (ten-frame, bundles/packs of ten, loose ones, total), but I probably could have used more.

Overall Reflection My overall average score was a 4.2. I feel pretty good about this score. I think that I am doing fairly well for a fellow in the middle of the year. I think that I have some strengths related to behavior management and providing feedback, but I need to work on incorporating students more and using different modalities, as well as giving my students more autonomy in the classroom.

Subjective Analysis of Teaching Episode There were a few aspects of my videotaped lesson that gave me some insight to my

developing skills as a teacher. One thing that surprised me was the way that I was able to use

questions to nudge my students toward finding the point of a lesson. While I want my students

to respond with what they truly notice in a lesson, there are still specific things that they should

notice that will help them build their skills. I always felt that I was not great at nudging them

towards the “correct” answer, but watching myself in the video showed me that I am

developing those skills. I was able to ask different questions to lead them to notice the

important aspects of a ten-frame. It was really interesting for me to see my own facial

expressions in the video. It was the first time that I have observed myself. I saw my “teacher”

look for the first time and was able to see the expressions I made when students were acting

out, giving insightful answers, and when I was watching students turn and talk. I noticed that I

tend to model good listening skills when my students are responding to questions. I tend to

give good eye contact and show signs of active listening. I also noticed how much movement

occurs on the carpet when they have been sitting too long. I need to reflect on and think about

how to incorporate shorter lessons or more movement into the activities to keep them

engaged.

I am not sure that the video shed any light on the students’ performance on the

assessment. I did introduce the language of bundling or packing materials into tens, the idea of

loose ones, and why we use ten as a friendly number. It could have laid the foundation for

students to understand the tens place and the ones place. However, I do not know that there is

any data from the video that supports any assessment data. This could be due to the fact that

the unit was not well aligned with the benchmark assessments, as mentioned earlier.

Course Take-Aways and Next Steps

I think that I have come a long way as a mathematics teacher. When I came into this

program, I was terrified of teaching math and I was not looking forward to taking on that task. I

had lots of bad experiences with math in my own academic career and I was anxious and

unsure about how to teach math, especially through a constructivist lens since I was never

taught that way growing up. I definitely believed in the idea of constructivist math, but I was

not sure how to teach constructivist math. Over the course of this class, I have been given

many resources about how to teach math in this new way. The math course was taught in a

constructivist manner, so I was able to have multiple experiences as a constructivist math

student. I was also grapping with this idea in my classroom, alongside my lead teacher and the

other first grade team. While we have had many unfortunate setbacks related to math in the

classroom, I think that we are finally getting our stride and we are doing our best to create a

constructivist math environment in our classroom. It is slow coming, but we are building that

foundation, and I feel more confident using this style of teaching than I was when I arrived in

July.

Moving forward, I need to continue to be reflective about teaching constructivist math.

How can I improve my lessons? How can I confer in a way that nudges students in the right

direction? How can I create authentic problem solving situations for my class? I also need

more coaching on how to write a strong math unit plan. This plan relied on the curriculum

provided to me by the school. I do not feel confident that I could plan an entire unit on my

own. I will need additional help and coaching on this as I move forward as a teacher. I also

need to accept that I will be grappling with this new concept of teaching math as my students

struggle with math itself. I need to accept that struggle and see it as a growing experience for

me. I look forward to looking back on this unit and reflection at the end of the year and next

year when I am a teacher-of-record. I hope I can look back and feel proud of how far I have

come as a mathematics teacher.

Appendix A: Place Value Assessment

Read the task below. Complete the task in the space provided. Clearly explain your thinking. You may use words, numbers, and pictures to make your explanation clear.

Task: Abby has a box of 28 crayons.

Draw a picture to show what the 2 in the number 28

represents.

Draw a picture to show what the 8 in the number 28 represents.

2 8

Explain the pictures you drew.

Appendix B: Place Value Assessment Read the task below. Complete the task in the space provided. Clearly explain your thinking. You may use words, numbers, and pictures to make your explanation clear.

Task: Sam has 45 pennies. His sister Ruby has 54 pennies. Who has more pennies? How do you know?Show your thinking.

Explain your thinking.

Appendix C: Place Value Assessment

Read the task below. Complete the task in the space provided. Clearly explain your thinking. You may use words, numbers, and pictures to make your explanation clear.

Task: There are 10 markers in each box. If you have 60 markers, how many boxes of markers do you have?How do you know?Show your thinking.

Explain your thinking.

Appendix D: Collecting Stamps Game Materials