What do Potato Chips and Line Dances have to do with Integers?

C. Mae Waugh Barrios

Final Course Reflection Assignment

Building Math Knowledge for Teaching Struggling Learners: Integer Standards (CCSS)

December 19, 2014

If you had asked me before June 2014 about chips, I would have replied, “Potato or tortilla?” Likewise, I thought a number line was something you did while square dancing. Just kidding. But I really had no idea of their significance in integer instruction and conceptual knowledge.

Before this course, all I knew about integers were the rules. I knew a negative times a negative equals a positive. I knew a negative plus a negative equals a negative. I knew to subtract integers, you just add the opposite. In fact, when I was in seventh grade, I remember our integer assignment was to write a narrative story with integers as the characters, in order to demonstrate that we knew the rules. I also remember getting a poor grade on the assignment, because my sixth-grade brain couldn’t keep the rules straight, because they really held no meaning for me.

I had heard about the chip model before our summer workshop, and even knew that yellow meant positive and red meant negative, but I had never used it before and I had never seen it in action. Completing operations using the chips and chip mats was completely new to me, but struggling through the process this summer was an enriching experience. What really struck me the most was how chips are able to help represent why the integer rules exist. This helped me make meaning of the rules I had half-wittedly memorized so many years ago.

I had used number lines before, but only to build foundation for integer value and relationship, to show that negatives were less than zero. I’d never used it for any operations. Actually the first time I saw the vector model for addition was when I was co-teaching a math class and we were doing practice MCAS questions. With my numeracy knowledge as an adult (and math teacher) I was able to figure out what the diagram meant, but I had never been explicitly taught the meaning behind it. As we went over it at the October workshop, I felt conflicted over the procedure of always starting the vectors at zero. I thought it was an additional step that complicated the process and might only confuse my students. But when I actually taught the vector model to my students, I found it was an essential step for students to differentiate between addition and subtraction models on the number line. 

However, the activity that resonated most with me over the entire workshop was the integer clothesline. While it doesn’t specifically relate to integer operations, having the numeracy understanding of the placement and relationship among integers and rational numbers is an integral part to building conceptual understanding for students. I used the integer clothesline for all of my middle school math ELD students and it was impactful in so many ways. First, it was kinesthetic and got even my most active students engaged and involved. Second, it was a form of pre-assessment, so that I was able to determine my students’ background knowledge and experience with integers, because they come from schools from all over the world, with different instructional styles and pacing. And finally, it was an activity that provided immediate feedback to the students, not only me but also my students’ peers. If a student began putting an integer in the wrong place, his classmates were the ones to help him revise his placement. 

These three integer models helped me connect the rules and concepts I had learned in isolation to a foundation of conceptual knowledge and they gave me resources so that I could help my students do the same. 

Obviously in order to successfully compute integer operations, students need number sense and operational fluency. They need to know how to add, subtract, multiply and divide multi-digit whole and partial numbers and they need to know what these operations actually mean. As I began teaching my students about multiplying integers, I asked them what 3x4 “meant” and all my students answered, “12.” They had memorized math facts, which is definitely important for computational fluency, but they were missing the conceptual foundations. If they didn’t know 3x4 meant 3 groups of 4, how could they model 3x4 using the chips? How could they model it using a number line? Additionally, using models helps students build the conceptual knowledge that subtraction is really addition of negative numbers. I told the students subtraction actually doesn’t exist and they couldn’t believe it! I challenged them to give me a subtraction problem that I couldn’t change into an addition problem, and of course they were unable to stump me. 

Having an understanding of integers and integer operations is an important foundation as the students move toward algebra and then on to calculus. 

Yet none of the models were a perfect method solitarily and the students struggled with difficulties and misconceptions with each one. All models have their limitations and their benefits. It was difficult for the students to transition from adding using the vector model to subtracting using the vector model. While the vector model helped to denote the directionality of each operation and integer, manipulating the vectors from one operation to the next was so confusing for the students. They asked me why the process was so different and it was hard to get them to understand that subtraction is really finding the distance between two integers on the number line and addition was combining their values. Without this differentiation, the sums and differences were neither accurate nor understood. 

The Walking the Number Line activity proved to help the students to model operations with vectors on the number line. The Walking the Number Line activity has the students assume the role of integer and they move in the negative and positive direction, adding and subtracting positive and negative numbers. It helps the student build the understanding of the directionality of the negative and positive numbers. These skills then transferred to students’ paper number line representations.

Additionally, I learned to make sure students understand one operation model completely, before moving on to the next operation. That way students feel competent and confident with addition, for example, before moving on to subtraction, so they are less likely to confuse the two models. 

Using models to build conceptual knowledge helps the students to build meaning for the reason why the mathematical rules work. Rules are there for convenience, not to take the place of actual understanding and knowledge. 

Project Summary and Reflection: Paper-Pencil Probes with Whole Class

I. Administer the Probe   

1. Which probe did you use?   Why? 

In the project planner, I wrote about administering Probe 1 and 2. I have since administered and used almost every probe in the tool box, as I created my own unit for my ELL class about integers, using all the materials from our course this summer. I found all the probes were really beneficial, an not just for targeting my instruction. This was the first time most of my students have been asked to explain their reasoning or thinking. They REALLY struggled with the difference between giving the correct answer and providing an explanation. When I began teaching them about multiplication of integers, I started by just having them explain what 3x4 MEANT and they all said “12!” It took quite a while for the students to realize I wasn’t looking for the answer, I was looking for the thinking.

All this being said, I would like to use this time to talk about Probes 6 and 7 because those were the ones whose analysis I used to target the most instruction during this unit.

2.  When in your sequence of instruction did you give the probe to students? 

I spent a week in November working on addition and subtraction of integers with the students. I started with using the chip model to represent integers and then moved into chip model for addition, followed by the chip model for subtraction. Then I introduced the vector model on the number line. Going through instruction, I found the students grasped the conceptual operations of integers much more while using the chips than the number line. I administered probe 6 and 7 on Friday of the week I taught both addition and subtraction in order to see how much the students had retained. 

II. Process the Results 

1. Sort the work samples to look for class patterns of understandings and difficulties.

Class Chart for Probe 4. Integer Addition & Subtraction: Will the result be positive or negative?

2.  How did you sort the student work samples?  What categories did you use?   Why?

First I ordered the papers by the number of problems students got right. Then I looked

at just the students who missed 0 or 1 problem. Those ones I analyzed to see what

method they had used (number line or chip) and then I looked to see if they had used

the method correctly. Then I looked at the students who missed between 6 and 8

questions. Those ones I sorted also by method and then correct reasoning or unclear

reasoning.

3. What did the work samples show about students’ understandings?  What kinds of successful approaches did they use? Were any approaches missing that you would like students to use?  

It seemed as though the students who used the chip method had the best understanding and the most success. Unfortunately, a few students used the chip method. I think if more had used that method, more students would have done better. I have the probes before teaching the students any “rules” so it was measuring specifically conceptual understanding.

4. What kinds of difficulties and misconceptions did students have?  What stood out for you?
After I gave the probes, I regretted giving them on the same day and I regretted teaching them addition, then subtraction, then giving both probes. Upon reflection, I think students would have demonstrated better knowledge if I had given probe 6 after teaching the concept of addition and if I had given probe 7 after teaching the concept of subtraction. I think the biggest difficulty students had was with the number line, because for addition, they needed to put two vectors one after the other, depending on the addends. For subtraction, they draw two vectors from zero, each the distance of the integers they are subtracting. They just didn’t have the fluency to be able to manipulate the number line and vectors and moving back and forth from negative to positive direction. 

5. What might be the reasons for these difficulties/misconceptions?  Consider issues that can be addressed through instruction.

As I said above, I think administering the probes at the end of the week really didn't set the students up for success. I wanted to see which model they would choose and if they were really able to add and subtract integers after a week of instruction, but they proved to be just confused. Additionally, I think they chose the number line model because it was what I had taught them most recently, not necessarily was the one they knew the best. 

6. What questions did the findings raise for you about your students’ understandings and difficulties/misconceptions?  What do you want to follow-up on?

This made me questions whether my students actually knew what addition of integers meant. I personally find the most meaning with the number line, but the difference between using it for addition and using it for subtraction really confused my students. 

III. Target Instruction to address identified needs   

Targeting Instruction Planner, Part C

1. Review probe findings (part B) and identify 1-2 math learning needs to target.  

What will you target?  Why?

I would really like to target how to model addition and subtraction using vectors on the number line. The students inability to demonstrate competency on this on the probes concerns me regarding their conceptual understanding. Additionally, I told the students they could write their explanation with any model they want, and the fact that they chose to use the number line but 8/12 students used it incorrectly .

Learning needs to target:

#1. I can model addition of integers on a number line

#2. I can model subtraction of integers on a number line

2. What specific math concepts, models, & processes will you focus on to address these learning needs?  Consider math goals & sequence of instruction

I will focus instruction on the operations of addition and subtraction, using the number line to model equations. 

3. Possible Next Steps.  What will you do?  Check all that apply.

__A. Move to Next Lesson with adaptations

                _X_Create one version for whole class          

__Differentiate for groups of students

3.1 What are your reasons for this choice?

The students clearly demonstrated that they do not understand the directionality of integers and that difference means the distance between two integers on the number line and addition means one vector leads to the next. 

4. Brainstorm Specific Ways to Target Your Instruction 

What activities, models, and strategies would help address the identified needs?

Consider ways to:  make adaptations, differentiate, add new content, and/or revisit prior content.   

Before adding and subtracting integers, I never did the “walk the number line” activity and I think it would be a great lesson to build the conceptional knowledge and directionality within the students. It also is kinesthetic, so it will reach tactile learners. Making the students act out the movement of the integers on the number line themselves might help them be able to draw it.

5. Write a Lesson Plan that describes what you will do.  You can either make adaptations to a lesson (from your math program or another source) or create a new lesson.   

• Lesson: Walking the Number Line, from our summer institute

IV. Implement the Lesson and Reflect on Instruction  (60 points)

1. Describe how you implemented your lesson.  What changes, if any, did you make from your original lesson plan for targeting instruction? 

At first, my students were as confused with the walking the number line game as I was this summer! Eventually after a few practice rounds, the students figured out the directionality piece and began helping each other. The students loved the role of spinner, of course, but the recorder was where the students really began to understand integer equations. The students became quite competitive and really got into the game. It also helped reinforce that the larger the negative number, the farther away from zero it is.

2. How helpful was the targeted instruction for your students?  

a) Circle your rating.

1             2 3 (4)         5

Not Helpful                                       Helpful Very Helpful

b) Give reasons for your rating.  In what ways was the instruction helpful or not helpful for targeting the identified needs of your students?  Provide sample student work or responses to support your explanation.  

I found the students really participated and interacted when we work working with the chip models. It must have been the tactile component. Therefore the number line vector model was difficult to grasp. Walking the number line was interactive and tactile, so that the students became the integers and moved along the number line themselves. Adding and subtracting opposites are dense concepts that can be difficult for students with little to no numeracy to really internalize. This lesson helped the students

3. What might you do differently if you use the strategies/activities again?  Why? 

I would definitely revise the student handouts and make them more friendly for ELD 1-2 students. The students struggled with the sentences, even on the “Walking the Number Line Reference Card” the English is extensive. Instead of giving it all as a hand-out, next time I might just explain the game to the students themselves and have them act them out and figure it out as they go along, which they ended up doing anyway.

V.  Overall Reflection on Project   

1. What are 2-3 things that stood out for you in the process of carrying out this project: administering a probe, processing results, and planning & teaching targeted instruction? 

Completing the probes was probably the first time most of my students have ever been asked to explain their thinking, especially in English and with diagrams. I used them in many ways—as pre-assessments, post-assessments, formative assessments and extra credit. At the beginning of the unit, I photocopied every single one, and the ones I didn’t administer, the students actually enjoyed doing as extra credit. After assessments, early finishers could do unused probes for extra practice. They would complete them and bring them to be and I would score them immediately, for instant feedback, letting students know not only if their answers were correct, but also if their explanation demonstrated conceptual knowledge. 

Administering probes as a form of formative assessment was so helpful to guiding my instruction. They were quick to grade and even when I didn’t go through the complete process as I did for this one, looking over my students’ shoulders as they took the probes, I was able to see where their deficits were. 

Although the students balked at first, taking the probes also challenged them in constructive ways. The motto of my class is, “Growing is hard and that’s okay.” These probes were an excellent opportunity for students to struggle with the material and give me a ton of data.

2. What challenges, if any, did you face in carrying out the project?  How did you address them? 

Through the project and my integers unit, I utilized almost all the resources and strategies I learned at the workshop. I have not felt as successful or had my students feel as successful as we did when I was creating and teaching this unit and the lessons myself. For the beginning of the school year, I was trying to follow the grade-level content and pacing guides and books to no avail. But the way in which these materials were sequenced really helped my students succeed. 

Unfortunately, I did have to spend a lot of time scaffolding the language aspects, because while the workshop was geared toward struggling learners, the needs of ELLs impact learning in different ways. I had to rewrite a lot of the worksheets to have less English and explanation, in favor of more problems and investigation.

Additionally, I got to the end of the unit I found I had worked so much on my students’ conceptual knowledge, they had little actual fluency for solving mixed integer operations. I ended up having to supplement the last week of instruction by creating a review packet that is much more practice-based.

3. What are two suggestions that you would give to future course participants on how to use probes with students, analyze probe findings, and/or target instruction?

In the future, I would tell course participants to not be afraid or overwhelmed of something new. In this workshop, I learned an entirely different way of interpreting and computing integers than I learned in school. I never worked with chips and never drew a vector on a number line. Sometimes I felt like I was learning right along with my students. However, having the conceptual knowledge really aids comprehension, especially because rules can be forgotten, but knowledge is forever.

Additionally, all the probes can be overwhelming, because they can feel like just more work for the teacher to do—more papers to grade. However, sometimes I found myself not even grading them, just glancing over them to see whether students “got it” or don’t. If it was a small group of “don’t,” I’d circle to them during class. If there were a lot of “don’t”s that meant a mini-lesson was necessary. The probes actually saved me time because they told me exactly where my students were at, so I didn’t send precious lesson time teaching them what they already knew; instead I could focus on exactly where their deficits were.