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In entry one, I am featuring my Pre-Algebra students. There are 53 students total, split between three classes. The students range from 13 to15 years old and are all in eighth grade. The population is composed of 33 males and 20 females. Twenty-five of the students are from a minority background, twenty-six students have either free or reduced pay lunch, and seventeen students are in the magnet program. Academically, my students are considered to be at or below, standards for annual yearly progress. Only 18 students met standards on our state test, the rest were below. Eighteen of the fifty-three students are in our special education program and twelve of the eighteen receive special services specifically for mathematics. Besides performance on the standardized tests, students in this group tend to be afraid of math and have not been successful in previous years. They come to me with the understanding that they are bad at math and that they will not get better. Due to this belief, they are not very motivated to complete work.
From a learning standpoint, the students prefer to do activities where they can work together and use manipulatives or technology to complete a task. Many are below grade level readers, so they prefer to learn by experimenting instead of taking notes or reading about new vocabulary. When we do have notes, the students prefer to just know the process opposed to why that process works. They struggle with any type of problems that require application or thinking beyond rote processes.
The subject matter I am focusing on is real-number operations, specifically integers. The students have learned the four basic operations with integers in the past, but they do not have a deep understanding. They know how to perform the operations by following the rules, but they do not know why those rules exist, or how to explain why the answer is what it is beyond listing the rule. They also get the rules confused when moving from adding and subtracting to multiplying and dividing.
This unit focuses on real number topic integers. The first enduring understanding that students should have at the end of the unit is that various models, such as drawings or manipulatives, can be used to explain the operations with integers. The second enduring understanding is that correct solutions require accurate computation. Students should be able to analyze any problem and explain why it is correct or incorrect. The final enduring understanding of the unit is that when problem solving, students should be able to select, apply, and explain the method for computing with integers.
These goals support students in their understanding of integers. These goals do not solely focus on the proper procedures for computing integers. Instead, they focus on the understanding of integers. They encourage students to use their number sense to find an answer, not just follow a rule. They also allow students the opportunity to think conceptually. By problem solving, students are using real world situations not just to compute, but to interpret integers. In addition, the goals also develop students’ critical thinking by giving them the opportunity to justify the accuracy of an answer.
For this unit, we engaged in several activities, all centered on using manipulatives as a tool to understand the computation of integers. On day one, we started with the absolute value of a number and used number lines to see that absolute value is about a distance, not direction. We used real life examples such as gaining and losing yards in a football game to relate the idea of absolute value to signed numbers. An idea related to absolute value we talked about is the magnitude of a number. For example, using counters, we can see that four red counters are more than two yellow counters, but we know that positive two is greater than negative four.
On day two we moved on to addition of integers. For this lesson I activated students’ prior knowledge of the number line by using a picture of an elevator, a vertical number line, to visually see what happens when we add positive and negative numbers. This activity is one that I am featuring in this entry. Instead of looking at both signs of numbers and developing “rules” for each situation, we looked at where we started on the vertical number line and which direction we moved based on the second number. We developed the patterns and then wrote the rule based on the patters. On day three, I continued the ideas of addition by introducing colored counters as a method to combine integers. With the colored counters, we were focused on the groups of positive and negative numbers and we talked about zero pairs, or opposites.
I used students’ prior knowledge of groups on day four when we discussed multiplication of integers. Students used colored counters in previous years to learn the idea of multiplication as repeated addition. In this activity, the Students used the colored counters to form positive and negative groups of numbers. By looking at the numbers and their signs, we developed groups of red or yellow to determine a product. This activity is also featured in the entry.
For my featured activities, I am focusing on the first enduring understanding: various models, such as drawings or manipulatives, can be used to explain the operations with integers. I chose to feature these two activities for two reasons. First, they are both visual models for students to see and manipulate the numbers. With the number line, they are able to understand what is happening when they are subtracting a number that is bigger than the first. They are also able to see how the magnitude of each number affects the outcome. This activity provided a way for them to understand signed numbers as a concept, not just as a set of rules to follow. The colored counters also give students the opportunity to understand why the two rules of integer multiplication exist. By making groups and looking at patterns, they can see why two negative numbers make a positive product. Most students can follow a list of procedures over and over again. What they do not walk away with, however, is the reasoning behind the rules. I want my students to understand why, so that they are able to apply what they have learned outside of my classroom. Part of this understanding comes from looking at patterns, which they do not have the opportunity to do if I present them with four rules to follow and when to follow them. By allowing the students the opportunity to manipulate the numbers themselves and see the pattern, they are developing habits of mind that will stay with them always. Both of these activities are visual models that students can use when explaining operations with integers.
An underlying understanding that comes out of this goal is developing patterns. Both activities help students develop patterns to understand the overarching rule. Students were able to see that mathematical rules and procedures are not just “magic”, but the result of looking at patterns and creating that rule or procedure. While working on the first activity, I had to really push the students to notice what was developing. When they discovered the pattern, it was then easy for them to tell me what the rule was and why. Putting them in that habit of mind prepared them better for the second activity. From the first example, they were trying to see the pattern, to make a connection between the rules that they had previously learned to what was actually happening on their paper.
For both activities, we worked together on several examples, each one a different case related to the rules of adding and multiplying. I asked the students during the activity to describe what was happening. I also encouraged them to compare the different examples and explain why the change in the number changed the outcome. During this time I was able to see if the students understood what the activity was intending. I was also able to clear up the common misconceptions as we went along. If students were struggling to see what was happening in a certain case, I was able to adjust the instruction and provide them with additional examples of that case. Independently, I assigned the students some reflection questions to complete. These questions required students to think about the patterns we developed and asked them to justify whether these patterns were always true. By reading their responses, I was able to see if my instruction was successful in allowing my students to meet the goals, or whether I needed to adjust it by re-teaching or correcting a misconception.
In teaching this mathematical idea, a major challenge for me is that many students prefer to apply a rule to solve a problem without understanding why. When I ask students to how they got their answer, they repeat back to me the rule that they wrote in their notes and on their paper. Students are typically able to explain accurately if I put a problem in a real life context (ex: if you go to the store with $10 and your bill is $12, what happens?). But, if I ask them why 10 subtracted by 12 is negative 2, they cannot make the connection, instead stating the rule. A challenge that presented in this class is that while some students knew the rules, some did not know them at all, or mixed them up and used the wrong rule (ex: negative three plus negative three is positive six, because two negatives make a positive). In the past, I have taught the rules, and then attempted to make connections and go deeper. Because of these challenges, this year I designed my instruction around only visual activities. We did not write down the rules and then practice examples. Instead, we used the visuals to develop the patterns and created general statements. During the activities, when students tried to explain their answer using the rules from previous math experiences, I challenged them to go back to the visuals and use the visuals to explain why the sum or product is what it is.
I chose student A because of the challenge he has given me this year. This young man is eager to learn, and works extremely hard. He is the type of student from the outside that every teacher wants. Unfortunately, as hard as he works math is still a major challenge for him. He is one of our special education students, qualified in math and behavior disorder. He is not afraid to ask questions or receive feedback. He will redo things multiple times if asked, and will not give up if he is encouraged. However, as often as I have re-taught a concept or taught it in a different way, he still does not always walk away with a strong understanding. He may even understand it well during class, but does not retain the information for homework. His goal is to always win: the only time I have seen him frustrated is when he does not win a game we are playing in class. In math class, he is happy if he gets the answer correct, even if he really doesn’t understand why. He is capable of applying an algorithm when given a problem, but he is not always capable of analyzing a problem to know which algorithm to apply.
When I wrote this unit plan, I knew that the students had been previously taught the rules of integers. One of my goals was for them to understand why the rules work, so that they did not confuse the rules, and could recreate the rule if they forgot. When looking at student A’s work, I see that I helped him make some gain in his understanding, but not as much as I wanted. When analyzing his response to the first instructional activity, I can see improvement. When given a pretest, student A simply added the numbers and made them all negative (ex: -2 + 3 = -5). By using the elevator, student A was able to visualize what was happening, and in his written explanation he showed understanding that adding means a positive movement on the number line. He also understood that adding will always do this. Unfortunately, the elevator was not as useful with subtraction. While student A recognized that subtracting means a negative movement on the number line, he did not correctly number his line in his third example, causing him to have an incorrect answer. This tells me that he understands the idea of movement, but is still struggling with his number sense. What he should have realized is that subtracting from a negative number results in a smaller number, when in fact his answer was bigger. This provided me feedback that I needed to spend more time with student A and possibly other students about checking for reasonableness of answers, and how the magnitude of numbers affects the outcome. After analyzing his entire response, my final assessment of activity 1 for student A is that he still has only a basic understanding of what is happening when performing addition or subtraction on signed numbers. However, he now has a tool he and I can both use as a gateway to deeper conversations about magnitude of quantities and direction. My goal was to move students away from dependency on rules, and I feel like I made progress toward this goal with student A.
When analyzing student A’s response to the second instructional activity, I felt he was more successful in meeting the learning goals. During our modeled instruction time (see Notes: Multiplying Integers) he successfully modeled the products when given the case of one positive and one negative. He was also able to create his own examples and find the product. A typical misconception when multiplying two negative numbers is that the product is negative, like when adding two negative numbers. When we worked on this part of the instruction, student A did originally have a negative product. I noticed this when students were working independently, and I was able to sit with him and have him think critically about his answer. I had him again set up his problem with the counters, and we talked about what the opposite of 5 groups of -3 would look like. After the feedback I provided him, he understood why the answer should be positive. He told me it was easier for him to think of the first number as simply “the number of groups” and the second number as “the color” (red or yellow). He said once he had his model set up, then he would go back and look at the first number to see if it was positive or negative. If it was negative, then he knew he needed to take the opposite of the product he currently had. I asked him why he liked this way of thinking, and he said because “it means I only have to think about one way. I don’t have to know a bunch of rules.” When looking at his homework (see Multiplying Integers Homework), I see that he understands the concept. However, his verbal explanation does not really show his level of understanding. When I asked him what he meant by “no because you haft to x” on question nine, he told me that since he knew how to multiply, he could just do that instead of the groups. I asked him how he would determine the sign of the product then, and he said he just looked to see if there was one negative or two.
When comparing both responses to the instructional activities, I feel like student A made gains in his conceptual understanding. By using the manipulatives he had the opportunity to think about what was really happening when adding and multiplying integers. He no longer has to rely on memorizing a set of rules and hopefully not mixing them up.
I chose student B because like student A, she too has been a challenge for me this year. Student B is also a hardworking student, but unlike student A has a much higher level of understanding. She is one of the 34% of students in her class that passed our state assessment, and she has maintained an “A” the entire year. She is a hardworking student, who completes all assignments without complaint, and is tenacious about getting to the correct answer. Student B is also a teacher pleaser, so she frequently participates in class discussion. Again, student B from the outside is the student that everyone wishes they had a classroom full of. Academically, she is a star performer. However, she has a very strong opinion that she is not good at math. In our district we have differentiated our classes by ability levels, and student B is in the lowest level of math that we offer at eighth grade. What she struggles to understand is that even though she is in what she refers to as the “dumb math” class, she is in pre-algebra, which was considered to be the grade level course for eighth graders until recent reform. She only sees that there are two math classes above her and none below. To further weaken her math self-confidence, she is in advanced language class and advanced science class, which makes her feel math is her worst subject. In working with her this year, I have made many efforts to encourage more positive math self-esteem.
Student B’s work for the first instructional activity tells me that I achieved my learning goals. Student B was able to successfully model addition of integers. She understood by looking at the manipulative that adding a positive number will always result in a forward or greater sum, and subtracting a positive number will result in a backward or smaller sum. She was able to do this without relying on a rule, but because she understood the direction and movement. Something interesting about her work is that she also addressed the “adding a negative” situation. Her problem statement to “start 2 floors below ground and move down three floors” was -2 + -3 instead of -2 – 3. Students typically have the prior knowledge that rewriting a subtraction problem is “adding the opposite of the next number” but they struggle with the converse of that (i.e. -2 + -3 is the same as -2 – 3). When I asked student B what led her to that conclusion, and she told me that “I just thought of it like adding a movement, even though it was going down. That’s why I put the negative.”
Her response to the second instructional activity provided me with some valuable feedback I will be talking about in my reflection. Student B was not as successful with the multiplication activity, so I don’t feel like I achieved my learning goal. Student B had shown mastery on this concept during her pre-test; where I’m positive she used the rules she had previously been taught. However, her work shows that she did not truly master the idea of negative meaning opposite. For problem two, she wrote in words “the opposite of -2 groups of 3” but her model showed two groups of negative three being flipped to positive three, instead of the reverse (two groups of positive three being flipped to negative three). Her model in this case led her to the incorrect answer. However, she successfully modeled (-2) x (-4). This tells me that she is good at referring to her notes, but does not trust her own prior knowledge. In question eight, she clearly explained that if the product of two numbers is negative, then one must be positive and one must be negative. However, that’s not what she did for questions two and four. This also tells me that she may not have a deep conceptual understanding, but is good at memorizing a rule. This could also mean that she is not using her number sense or checking for the reasonableness of her answer. Either way, I did not meet all my instructional goals in her case.
Overall, I feel like these two instructional activities helped student B think about integers as a concept and not a set of rules, but the activities alone were not enough to build a strong conceptual understanding. When I handed back the “Multiplying Integers Homework,” I asked student B about the feedback I gave her on question eight. She looked at her response to question eight, and then looked at questions two and four. She became very perplexed, so we pulled out the counters again, and I had her model what she had written. Working with her, she was able to see her error. I reminded her to always check for the reasonableness of her answer. She knew what the outcome should have been, but she did not stop long enough to think about it.
In general, the work of these students suggests that I need to look at how I model. I need to think carefully about how to connect the models to conceptual understanding. By modeling students are able to see what is happening, but I think that sometimes they are still just following a procedure and drawing pictures, without building a deep understanding. I need to work on the questions I ask them while we are modeling in order to pull it all together.
After reviewing student responses to activity 1, I would make a few changes. Before teaching activity 1, I will reteach students how to create a number line, or provide them with an already numbered number line. I assumed that my students had mastered this bit of prior knowledge, and it ended up confusing some of my students, leading them to incorrect conjectures. Also, thanks to student B, I will be adding problems similar to “-2 + -3.” This will give me the opportunity to build a stronger connection to the fact that -2 + -3 is the same as -2 – 3. Finally, I want to build in more discussion about connections so they can see why the rules they were previously taught really work. I did a different activity prior to this one that did make a connection to the rules, so I need to find a way to connect them all together. I will repeat the concept of moving up and down the elevator in the future. My students have referred to the elevator many times since we have completed the activity. While they do not draw the number line anymore, they will explain “starting two floors below ground and moving up three is …” when they are asked to explain their thinking.
After reviewing student responses to activity 2, I will be repeating it in the future. This activity made a conceptual connection to the rules they had previously learned. I will be teaching it in a different manor, however. First of all, I will spend more time emphasizing that negative means the opposite of positive before we get to this activity. We talk about positive and negative being opposites, but not specifically that negative three is the opposite of positive three. I will continue using this language when I model the first set of problems. Instead of three groups of negative four, I will say three groups of the opposite of four. The reason for the change is an attempt to avoid the confusion of problems such as (-2) x 3. Also, I will require the students provide a written explanation of how they connected the activity with the counters to the rules. We verbally discussed the connection as a whole class, but I would rather give the students an opportunity to think about it on their own.
For both activities, I plan to provide more opportunities for small group talk time including time for whole group share-out. I did a lot of modeling up front with both activities, and I would like to see if I can challenge them to build the models on their own in the future. This will help me see if they are using the models to understand the concept, or just following a procedure once again that they may not understand why they are doing it.
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