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Daily Lesson Plan: Mathematics Lesson (used in class)

Overall lesson topic/title: Problem Solving Mathematical Task: Today’s Special Number

###### Goals/Objectives/Big Ideas:

• Students will explore number composition and part-whole relationships (example: 10 can be 5+5 or 20-10)
• Students will begin to get used to working with equivalent arithmetical expressions
• Students will use problem-solving skills to work with different operations to get to the same number solution
• Students will work in groups to encourage discussion and interaction to help each other find ways of thinking.

N.FL.02.06 Decompose 100 into addition pairs, e.g., 99 + 1, 98 + 2…  *(using 10 instead of 100 for this first lesson)

N.MR.02.08 Find missing values in open sentences, e.g., 42 + ■ = 57; use relationship between addition and subtraction.

Materials & supplies needed:

•    Large pieces of paper for board (with magnets)
•    Markers
•    Pencil and paper per student
•     Manipulatives (blocks)- if needed
•   100 chart per group

-Previous Knowledge: Addition, subtraction, parts of an equation, number order, equivalent equations, skip counting.

-Possible Student Methods/Ways of Thinking:

• Students begin with commonalities such as 5+5, 10+0, 10-0
• Starting from 10+0, students would subtract 1 from the 10 side and add 1 to the 0 side, while seeing that addition has the commutative property.

Ex.)  10+0                     4+6

9+1                      3+7

8+2                      2+8

7+3                      1+9

6+4                      0+10

5+5

• Starting from 10-0, students will add 1 from the 10 side and add one to the 0 side

Ex.)  10-0

11-1

12-2

13-3, etc.

• Students will notice that using the same number in the ones place to subtract will give 10 if the tens place is a 1. ex.) 1# - # = 10 or 16-6=10 (where 6 is the #)
• Students will select a number between 0-9 and subtract that number from 10 to get what to add. (ex- students selects the number 3, then does 10-3=7, then uses 3+7=10)
• Students will subtract numbers from 100 that are in increments of 10 (easy and familiar to work with) ex- 100-90=10, 60-50=10
• Students will right down all number adding to 10, using 0-9 digits ,then automatically write down their commutative pairs. (ex- student writes down 6+4 then writes 4+6)
• Students will use their 100 chart to visually see which numbers are 10 apart (ex.- students see that 60 is 10 apart from 50, so student will know 60-50 is 10)

-Student Misconceptions/Errors: addition is not commutative, subtraction can have a smaller number before the larger number without getting a negative number, you can only use 0 to 9 to get 10, students may use same numbers to add and subtract because they may believe it works for both, students may use 10 in the integer position instead of the solution position.

Introduction of Lesson:

-Introduce lesson as “Today’s Special Number” à the number 10!

-Tell students they will be working with addition and subtraction to get to the solution or answer of 10

-Do a warm-up activity on the board to get students adding and subtracting (5 problems together) and go over parts of a number sentence so students know that 10 is the SOLUTION and they are looking for the two other parts of the number sentence

à 15+5         9+8           25-10       90-20    7+2

-Tell students they are the people who get to make the number sentence (investigators!)

-Refresh student memory of what a 100 chart is and how they skip counted before (counting down the line of numbers)

-Give an example of one of the solutions to get to 10: (5+5) to give students a starting point to work with and understanding of what a solution would look like. 5+5=10

-Tell students they are going to work in groups to find as many solutions as possible to get to the number 10, and that they are to work/think together as a group: they are allowed/encouraged to talk.

-Separate students into groups of 4, making sure each group has paper and pencils for each student to work with

-Circulate around the room, observing each group and picking out interesting comments or solutions to discuss after task is completed.

-Does the number order matter in addition sentence?

-How did you decide to do the problem in this way?

-What do you notice about adding up to 10 that is different from sub. to get to 10?

-How else could you start/solve this problem?

-Once about 10-15 minutes (?) has passed, elect one student per group to go up to the board to write their solutions on the large piece of paper with a marker.

-Go over certain answers with entire class, discussing interesting solutions and ways of thinking- have students come up and show how they found out their answers on overhead or board or through oral discussion. (choose solutions/ways of thinking that cover various ways of how to get solutions while circulating around the room- make note).

-Important “after” questions:

-Who can explain what (STUDENT) did?

-What did we notice when we were using subtraction to get to 10?

-Are there different ways of doing this problem?

-After going over 2-3 different ways of going about the problem- finish off by reminding students that math problems may have different ways of solving/thinking, but that the solutions can still be correct.

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#### Project 3: Part I

• What are your goals for the lesson?  What mathematical content and processes do you hope students will learn from their work on this task?

-Goals/Big Ideas for the lesson include:

Students will explore number composition and part-whole relationships (example: 10 can be 5+5 or 20-10)

Students will begin to get used to working with equivalent arithmetical expressions

Students will use problem-solving skills to work with different operations to get to the same number solution

Students will work in groups to encourage discussion and interaction to help each other find ways of thinking.

-The GLCE’s being focused on include:

-N.FL.02.06 Decompose 100 into addition pairs, e.g., 99 + 1, 98 + 2…  *(using 10 instead of 100 for this first lesson)

-N.MR.02.08 Find missing values in open sentences, e.g., 42 + ■ = 57; use relationship

- The overall math process I want the students to learn from working on this task is using problem solving strategies and ways of thinking to explore various ways to use addition and subtraction to find the number 10, and to understand how different number sentences work to achieve this solution. Also, I want students to begin to recognize patterns in solving a problem like this and I am interested in seeing the different starting points of how students go about this problem.

• In what ways does the task build on students’ previous knowledge?  What definitions, concepts, or ideas do students need to know in order to begin to work on the task?

- Student’s previous knowledge that they will need to refer back to includes addition, subtraction, parts of an equation, number order, equivalent equations, and skip counting. Addition and subtraction are asked of the students in order to get to the number 10. Parts of an equation, including what a solution is, is important because the students must understand that the 10 they are striving for is actually the solution and that they are searching for the other 2 numbers within the number sentence.

• What are all the ways the task can be solved?

-Possible Student Methods/Ways of Thinking:

1.)    Students begin with commonalities such as 5+5, 10+0, 10-0

2.)    Starting from 10+0, students would subtract 1 from the 10 side and add 1 to the 0 side, while seeing that addition has the commutative property.

Ex.)  10+0                     4+6

9+1                      3+7

8+2                      2+8

7+3                      1+9

6+4                      0+10

5+5

3.)    Starting from 10-0, students will add 1 from the 10 side and add one to the 0 side

Ex.)  10-0

11-1

12-2

13-3, etc.

4.)    Students will notice that using the same number in the ones place to subtract will give 10 if the tens place is a 1. ex.) 1# - # = 10 or 16-6=10 (where 6 is the #)

5.)    Students will select a number between 0-9 and subtract that number from 10 to get what to add. (ex- students selects the number 3, then does 10-3=7, then uses 3+7=10)

6.)    Students will subtract numbers from 100 that are in increments of 10 (easy and familiar to work with) ex- 100-90=10, 60-50=10

7.)    Students will right down all number adding to 10, using 0-9 digits ,then automatically write down their commutative pairs. (ex- student writes down 6+4 then writes 4+6)

8.)    Students will use their 100 chart to visually see which numbers are 10 apart (ex.- students see that 60 is 10 apart from 50, so student will know 60-50 is 10)

• Which of these methods do you think your students will use?

-I believe the students will most likely use method #4 because the pattern, if recognized, will produce many solutions. Also, method #1 will be common due to their familiarity with those particular equations.

• What misconceptions/errors might students have/make?

- addition is not commutative, subtraction can have a smaller number before the larger number without getting a negative number, you can only use 0 to 9 to get 10, students may use same numbers to add and subtract because they may believe it works for both, students may use 10 in the integer position instead of the solution position.

#### Planning the lesson

Supporting Students’ Exploration of the Task

• How will you ensure that students remain engaged in the task?
• What will you do if a student does not know how to begin to solve the task?

-Ask student if they know any equations off of the top of their head to get to 10 (ask about familiar addition/subtraction sentences leading to ten- method #1.) Also, offer blocks for visual representation of what 10 looks like and physically separating the blocks into groups of 2.

• What will you do if a student finishes the task almost immediately and becomes bored or disruptive?

-If student is finished early, I will prompt them to start on the number 20, or if the student is advanced, I will ask student to try to come up with number sentences that use BOTH addition and subtraction (ex- 5+8-3=10). Basically, make another challenge for them to work on. Also, I will prompt them to try to figure out a pattern on how to get a lot of solutions, or ask them to try a different method to thinking about the problem.

• What will you do if students focus on non-mathematical aspects of the activity (e.g., spend most of their time making a beautiful poster of their work)?

-I will go to their group and begin by asking them questions and ask them how they went about thinking about the problem. I will also ask them to come up with what they think is the best way to get the most solutions. I will tell them that I expect them to share their ways of thinking with their classmates after the task is completed, so they should be prepared for that discussion. Having the students understand that they are expected to prove their work will get them working on their work and stay focused (assessment expectations).

• What are your expectations for students as they work on and complete this task?
• What resources or tools will students have to use in their work?

• Large pieces of paper for board (with magnets)
• Markers
• Pencil and paper per student
• Manipulatives (blocks)- if needed
• 100 chart per group
• How will the students work -- independently, in small groups, or in pairs -- to explore this task?  How long will they work individually or in small groups/pairs?  Will students be partnered in a specific way?  If so, in what way?

-The students will in whole group during the warm-up and introduction of the task, split into groups of 4 during the task, and come back together as a whole group during the discussion of the task. They are able to work in small groups for 15 minutes. Students will be grouped according to math levels- creating a balance between groups, meaning each group will include average, advanced, and lower-level math thinkers in order to allow each group to help one another and work at similar levels.

• How will students record and report their work?

-Each group will write down their work and solutions on paper, then report/write their solutions on the large piece of paper at the board.

• How will you introduce students to the activity so as not to reduce the demands of the task?  What will you hear that lets you know students understand the task?

-To introduce the task, I will tell them they will be working with addition and subtraction, and that there is a special number today that will always be the solution of the number sequence- the number 10. I will warm-up their addition and subtraction thinking with 5 different problems (not resulting in a 10), and also give a simple example of what a solution would look like to get to 10, using 5+5=10. I wil tell the students that they are the investigators to find as many number sentences as possible to get the number 10 as the solution. To know they understand, I would want to hear them referring back to that 10 must be at the end of the number sentence (in the solution spot), agreeing or disagreeing with one another about if 2 numbers when added or subtracted come out with too much or too little (over ten or under ten), and also if the students are seeing patterns in the addition sentences being commutative or using the 100 chart to visualize the distances between numbers being 10 spaces. (example- the student notices that there are 10 spots between 50 and 40, so those 2 numbers can be used to get 10 with subtraction).

• As students are working independently or in small groups:
• What questions will you ask to focus their thinking?

-If using blocks- can you separate the 10 blocks into 2 different size groups?

-Ask them to pick a number from 1-10. From there, ask what you have to add to that number to get 10. (ex- student chooses 6. I ask “What do you have to add to 6 in oer to get the number 10?)

-What happens if I start with the number 7 and have to add to get to 10?

• What will you see or hear that lets you know how students are thinking about the mathematical ideas?

-Hear the concepts of addition order not mattering, number of spaces between numbers on the 100 chart, they can use numbers larger than 10 in the subtraction sentences, that 10 has more subtraction sentences than addition sentences.

• What questions will you ask to assess students’ understanding of key mathematical ideas, problem solving strategies, or the representations?

-Does the number order matter in an addition sentence?

-Why did you start/do the problem this way?

-Why are there more subtraction sentences than addition sentences?

-What do you notice about the numbers on the hundred chart that are 10 apart?

• What questions will you ask to advance students’ understanding of the mathematical ideas?

-What do you notice about adding up to 10 that is different from subtracting to get to 10?

-What do you notice about the addition sentence numbers? Why do they have to be lower than 10?

-Why does one number in the subtraction sentence always have to be equal to or larger than 10?

-What other ways can you start/solve this problem?

-What happens if we use the number 20 instead of 10?

• What questions will you ask to encourage students to share their thinking with others or to assess their understanding of their peer’s ideas?

-What do you think of (STUDENT)’s idea/way to solve the problem?

-How many of you solved the problem this way? Who solved it differently?

-Tell them I like their way of thinking and I want them to share this method with the class during discussion.

-Can you relate your idea to (STUDENT)’s idea?

Share and Discuss

• Which solution paths do you want to have shared during the class discussion in order to accomplish the goals for the lesson?

-I will want to share at least 3 different ways of looking at the problem. It is not the solutions that really matter, but the way of thinking that matters. I want to include a visual representation, a number representation, and a student’s thinking using the hundred chart. If any students come up with the patterns, I will definitely want to include those in discussions.

• Which will be shared first, second, etc.?  Why?

-The first would be the number representation, then visual, then hundred chart, then pattern. I would want to start with the numbers to introduce the concept, then venture into visual and hundred chart to allow students to see where the numbers came from. The pattern would be last because it is most difficult to understand and also, the students will need more discussion prior to seeing the pattern.

• In what ways will the order of the solution paths help students make connections between the strategies and mathematical ideas?

-Starting with numbers, the students can see what formats they are used to in addition/subtraction sentences. From there using visual representation or hundred chart will allow those numbers to be connected to the actual concept, and allow visual learners to give the number’s substance. The math ideas may be similar throughout, but the strategies will be different. The variety of strategies will allow connections to be made from the numbers on the page to an actual visual. Patterns in strategies can be seen then after students are exposed to both the numbers and what they represent.

• What will you see or hear that lets you know that students in the class understand the mathematical ideas or problem-solving strategies that are being shared?

-If students ask questions about other strategies, then that shows me they want to understand.

-If students can compare strategies and see any similarities between them.

-If students can explain other students’ ways of thinking.

-If students make comments about how they saw it differently but came up with the same solutions.

• How will you orchestrate the class discussion so that students:
• make sense of the mathematical ideas being shared?

-refer back to the connection between visual representation and number representation so there are various ways of understanding the concept

-add method of my own to give discussion balance

-scaffold students with questions during discussion.

• expand on, debate, and question the solutions being shared?

-ask if the solutions on the board all add/subtract to get 10?

-ask if they have any questions for the person sharing their solution

-ask if you can use (STUDENT 1)’s method to get the same answers as (STUDENT 2) got

• make connections between their solution strategy and the one shared?

-ask if any similarities exist between strategies

-see if solutions are the same or if any are different

- take a quick poll of who used what strategy and if they can add to any strategy on the board

• look for patterns and form generalizations?

-ask students to share any pattern that they discovered (if any)

-can we use the same method on a different “today’s number” solution?

-save the pattern discussion for the end when students are engaged in other strategies and have been exposed to different angles of viewing the problem they may not have thought about before.

• What extensions to the task will you pose that will help students look for patterns, make connections or form a generalization?

-Ask hypothetical questions about what if “today’s number” was smaller or bigger than 10, and if we could use the same methods to find solutions.

Project 3, Part II: Teach- What Happened?

(a) Teaching the lesson Today’s Special Number to my second graders both surprised me and allowed me to see what we learn in TE 401 put into action by the students. Things were lower, met, or exceeded my expectations about the lesson. Before the lesson, I dropped a lot of items I was planning on doing due to time problems. Since my school is a reading first school, little time is spent with mathematics. I was planning on introducing the lesson, having students do a warm up with addition and subtractions problems, remind them of what a 100 chart was, then give an example of a number sentence that had the answer 10 (which they were to look for as many as they could). What did occur was the students were placed in groups of 4-5 with their desks in the group formation. This excited the students because this setting encouraged conversation and was something new for them. To start the lesson, I had to first quiet the class down, which took a minute or two for me to have their attention toward the chalkboard. After they settled, I told them that the lesson we were going to do today involved us working together to find the answer to my question, and that they must work as a group in order to do this investigation. I then told them that the lesson was called “Today’s Special Number” and wrote the number 10 on the chalkboard. The students seemed to be anxious to start working in groups. Due to time conflicts, I did not get to warm up their addition and subtraction skills with a few problems. Instead, I led straight into what the lesson entailed. I told the students that they needed to work together to find as many number sentences as they could that had the answer 10. I proceeded to pass out the 100 chart while reminding them what it was (numbers in order from 1-100) and told them they could use this chart if they wanted to. Then, once the 100 chart was passed out, I gave them an example of what one of the solutions might look like, writing 5 + 5 = 10 on the board. I asked the entire class if they could think of one more number sentence that would add/subtract to get 10. Daniel answered, “4 +1.” I asked him to add that with his fingers, and he realized it was 5. Then another student yelled out “7 + 3 =10.” I wrote that solution on the board and asked the class in general if they agreed. Once they agreed, I sent the students on their way within their groups to find as many number sentences as they could to get the answer 10. This “BEFORE” portion of my lesson went fairly well, but I found it hard to control the students’ excitement to be working in groups, since they are not used to it. But, once I got their attention on the board, the majority of students was actively participating and engaged in the introduction of the lesson. They seemed to understand what I was asking for and were anxious to begin, so I allowed them to begin.

During the lesson, I walked around to 2 of the groups, while Sarah (my classroom partner) worked with the other 2 groups. While circulating between these 2 groups, I noticed many conversations and ideas that were exciting to me as a teacher. In the first group, I was initially hearing them try to use their 100 chart in order to find a starting place. This first group started off with 9 +1=10, then ventured into using 8 +2 =10. This student’s method was looking at the number chart and counting spaces between that number and the number 10 on the chart. For example, this student started with 9 then realized that 10 was one space away on the number chart, so he wrote down 9+1=10. I then asked him how he came up with that number sentence and he replied, “I counted on my number chart in the order of the numbers until 10.” I asked him to show me with his finger what he means and he began to count from 9 to 10 moving his finger from the 9 to the 10 and explaining how there is one space until reaching 10. I asked him to try another number with that same method so he went from 8 to 10, using his finger to count 2 spaces away. I asked him to show his group what he did. I was excited to see that the 100 chart was helpful for the visual learners in the class, allowing them to actually see where 10 is amidst the rest of the numbers. Before I left that group to work, I asked them to try to figure out some subtraction sentences. I then walked to the second group I was working with to see what discussions were arising there.

In the second group, there was arguing about people working individually instead of as a group, so I intervened and began to ask prompting questions. To start I asked Christine why she wrote 10-0=10. She said that if you take away nothing, then you get 10. I asked her why this is so, and she responded, “because if you take away 0, you get the number you started with.” I then looked at Casey’s paper and noticed hers said 11-1=10. I asked her how she got that answer and she said she counted back from the number chart, as the first student did in Group 1. Then, an interesting comment came about from Hunter. With 7 + 3=10 written on his paper, he asked me if he can use the “turn-arounds.” I asked him to explain what he meant by “turn-arounds” and he said the turn-around of 7+3is 3+7. to me, I was happy to hear this comment because it was one of my possible solutions I thought students would discover. I asked him if 7+3=10, and he agreed. Then I asked him if 3+7=10 and he also agreed, so he knew from there he was allowed to use those turn-arounds.

During the lesson and circulation, I was very pleased with student thinking, but at the same time disappointed in the actual group work. Most students were working either on their own or with a partner instead of as a unit of 4-5 students. My prompt questions written in my lesson plan were not all used, but instead I improved what I was going to ask by what the students were producing. They all had good ideas, but were having trouble sharing them within their groups. I then decided it was time to come back together as a class to start thinking about different methods. I had chosen 2 student examples to share with the entire class which were Hunter’s “turn-arounds” and Curtis’s 100 chart counting method. I wanted to start with the 100 chart counting method since it seemed like a good visual to come back to and the majority of students were trying to use this chart.

Overall, after the lesson was given and worked on, the students came up with many different ideas, made some important discoveries about addition/subtraction rules, and even made generalizations. I didn’t expect the students to work that well in groups because they were not used to it, especially in math. The students leaned toward working individually and some became frustrated because they seemed left out of the conversations, or the speed of the solutions was going too fast. I didn’t expect the students to get this frustrated with the lesson. Overall, I thought the lesson content was appropriate for this open-ended discussion and allowed for many methods to be used. What was hard was the time constraint and getting the students to get focused on the lesson at hand, due to their excitement to be in groups. The easiest part of teaching this lesson was prompting ideas out of the students while and after they worked on the lesson. This particular group of students seems to really enjoy math and have creative ways in thinking about numbers in general.

(b) Student thinking in this particular lesson was one of creativity, diversity, and innovation. At first, students began to pull out random numbers below 10 to have a starting point into get that solution of 10. Many started off with 9+1=10 because it was easy to understand. From there, many students used their 100 chart to see what other numbers to try to use. From there, methods began to emerge and patterns were noticed. Some students kept using the number chart to physically see what numbers to use, while others began to use zero, and others began to notice subtraction patterns such as whatever number is in the ones place, is the number to subtract to get 10 (ex- 13-3=10, because 3 is in the ones place in 13). Two students in particular, Hunter and Curtis, stuck out in my mind as students who dug into the lesson. First, Curtis’s method was one of using the 100 chart as a visual. After I started the class to work on the solutions, Curtis jumped right into his 100 chart. He circled the number 10 on his 100 chart because he knew that was the goal he was working toward. By circling 10, Curtis was making a noticeable reminder about what the constant was in the number sentences and gave him a visual to what the solution had to always be. From there, he wrote down 9+1=10. After asking him how he did it, Curtis shared with me that the number of spaces away from 10 is the number to add. From there, he continued with finding addition sentences with 6+4 and 8 +2, using this same method of physically counting the number of spaces between the first number in the number sentence and 10. His way of thinking was using a visual to SEE the difference or sum of 2 numbers and if those 2 numbers can lead him to 10 as the solution. From there, Curtis began finding subtraction sentences by allowing himself to use numbers beyond 10, such as 11-1, still using the “spaces between” way of thinking as his method.

Hunter came up with an interesting method/way of thinking when I observed his solutions. He realized that addition has the commutative property, which would double his solutions he had on his paper. He asked me if he was allowed to use this property, so that shows me he already knew this property with his given label of being a “turn-around.” This method or way of thinking was a window into Hunter’s prior knowledge and allowed me to realize that this property may be both known and unknown to the other classmates.

Another student, Aaron, had another interesting way to think about some of the generalizations made about subtraction sentences during the whole-class post-lesson discussion. After revealing Hunter’s “turn-around” method, I then asked the student if this would work with a subtraction sentence. Most students replied, “yes” so I definitely went on with this question. By giving the example 13-3 and 3-13, the students realized that this property wasn’t working for subtraction. Aaron then revealed that this number sentence would give a negative. I wasn’t sure if he really knew what this negative number really meant so I asked for an explanation and received, “they are lower than 0.” So far, I could tell Aaron had a decent grasp on what a negative number was, but the rest of the class seemed a little confused by this vocabulary. I decided to tap into this concept of negative numbers in asking if anyone could explain what they are. To my surprise, Daniel used the 100 chart to explain Aaron’s negative number concept. By Daniel using Curtis’s method to begin to understand what a negative number was, I noticed that these 2nd graders were capable of connecting methods without even being asked to. Daniel might have not known what a negative number was, but after Aaron said “lower than 0” Daniel could go back to this 100 chart visual and see where the numbers starting from 3, would go off the chart when subtracting 13. I believe using this number chart allows for better student understanding of where these numbers stand and the relationships between them when adding and subtracting. With Daniel using Curtis’s method, this proves that change occurred in their way of thinking, because before Daniel did not use the 100 chart to get his solutions, but used it as a reference after realizing how someone could use it when adding/subtracting. Students were beginning to bounce ideas off of one another and actually use each other’s ideas to understand the addition and subtraction sentence properties in a better way.

(a) The element of classroom culture I focused on in the previous project was communication between students, communication between student-teacher, and the type of answers/questions that students provide. Basically, I focused on what communication students develop during a lesson or task. Within this lesson, the communication was different than “normal” or routine math talk had been in the past for this class. Before, students were to work on math individually without talking at all, and the communication between teacher and student remained on a surface Q and A session with recall questions. In this lesson, I tried to introduce the opposite to the students, allowing and encouraging a lot of discussion amongst themselves and with myself, asking questions and thinking aloud. When given the opportunity, student communication can lead to important math thinking, and students that were so used to answering obvious questions were now able to think beyond what they have in front of them on paper. With this open-ended, multi-solution task, students had to communicate with one another to start thinking about how to go about solving the problem. One thing I learned about student communication with second graders was that when working together, they seemed to need more structure within the group instead of allowing them to go ahead and work together as a unit of 4-5. This is one thing I would do differently in my lesson: give certain students particular roles within the group such as a facilitator, recorder, etc. Because of their age and what they are used to, these students needed more of a focused path in what they were supposed to do. Even with the frustration of not being used to working together, the students bounced ideas off of one another after sharing methods and answers. The student-student communication, when led by the teacher, allowed for students who were unable to start the problem to eventually find their way and get into a mode in which the task seemed easier than it seemed at the beginning. The students who led the way with thinking of methods immediately were encouraged because they were able to share their ideas and have to explain these methods with the entire group or class.

The communication between student and teacher was also different than before. Before, the teacher would ask questions with one particular answer (example: Which place value is the 2 in the number 342?). I tried to ask more prompting questions to lead them into methods they were half-way toward. For example, when Hunter brought up the idea of “turn-arounds” with addition, I asked him if he noticed a pattern of which number sentences could do that. Later, he understood that subtraction isn’t allowed to switch numbers. By prompting students into extended thinking, the communication between student and teacher takes a more significant role and leads the student toward even more discovery.

Overall, the lesson and task went well for this class’s first experience with math group work, but there are things I would do differently if I did this same math task with this same class. As I said before, I would give each student a specific role within his or her group so there is more structure to rely on. During the task, students were working alone or going too fast for some students to comprehend. By giving out roles such as facilitator, recorder, speaker, the students could focus more on the task at hand and know who is in charge of what. Another thing I would do differently would be to make sure I would have enough time allotted for me to give a warm-up, and allow for more methods to be shared to the class after the task was accomplished. I felt rushed because of the CT’s obligations that I had to pick and choose what to do within my lesson plan. A third thing I would do differently would be to get more students involved in the post-lesson discussion. I found myself calling on a limited group of students who I knew had good methods because I heard them during my circulation. Next time, I would want to get more students involved and speaking during the discussion so I can get those students who were not fully participating during the task, to begin thinking during the discussion. Getting each student involved is important because each student can then take something out of the lesson. Even if a student would just learn how to use the 100 chart would be an important skill to get from this lesson.

(b) The students received many things from this lesson. They either refreshed their memories or learned that addition is subject to the commutative property, learned how to use the 100 chart when adding and subtracting, learned to check their work to make sure the solution was 10, and gained experience with equivalent number sentences (solution is constant while the first 2 numbers change). Therefore, a follow-up lesson should touch on each of these aspects. A follow-up lesson to this task could involve the students receive a list of number sentences all containing the solution of 25. Some of the number sentences would not really add/subtract to 25, some would come out with negative numbers, and some would be correct. The students would work in partners to decide whether these number sentences are either correct or incorrect and they must be ready to discuss why they are either correct or incorrect. After this task is finished, we would come back as a whole-group to discuss what we think and why. This follow-up lesson would give students an opportunity to use the 100 chart again, decide which number sentences are equivalent, work with a “harder” solution than 10, and understand why the subtraction sentences cannot have the commutative property. The exercise would be in discussion format once again in order for students to help one another, come up with even more methods and to bounce ideas off of one another as to why or why not the number sentences are correct. An example of the worksheet would look like this:

Follow-Up Task to Today’s Special Number

Directions: Look at each number sentence and decide if it is correct or incorrect. If it is incorrect, explain why. Show all work on the paper and write how you decided if it was correct or incorrect: (what method did you use?)

1. 100-75=25
2. 0-25 =25
3. 2+5=25
4. 6+19 =25
5. 50-25 =25
6. 99-74=25
7. 20 +5=25
8. 11+14 =25
9. 14+11=25
10. 2-23=25

This task is practice in the new lessons learned in the first task, allows students to exercise their new knowledge, and allows for methods to be put to the test. In using 25, the solution is more difficult to work with because it is a higher number and also an odd number. When using the number 10, students had a lower amount of possible numbers to add together to get 10. With using 25, students must go beyond the single digits and begin thinking about adding double digits and working with odd numbers. Also, students may find the fact that 2 odd numbers when added create an even. Having another generalization “behind-the-scenes” of the task allows students to discover yet another rule, which is building their knowledge about addition and subtraction sentences. After the task was given, during the discussion I would teach the class about adding and subtracting even and odd numbers. I would lead them into patterns about what kind of numbers E+E, E+O, E-E, E-O, O-E give as the solutions.

Some students may have trouble with this new task and concept during this follow-up lesson. Some student problems could include: finding 25 too difficult to work with, not remembering or understanding how the 100 chart could be used to check these solutions, trouble with adding/subtracting double digits, and having the misconception that subtraction can have the commutative property. To help these students with these troubles I would have them work with certain people that understand how to use the 100 chart (make sure each pair of students has a student capable of working with the 100chart fluently). By starting with this 100 chart, the student could have a visual of what numbers they are working with. If students are having trouble with double digits, the 100 chart could be helpful in order to have the students physically count the spaces to see if there are 25 in between them. Another way to help these troubled students would be to use manipulatives, such as blocks in order to have something physical to rely on. The students would be able to represent the first 2 numbers in the number sentence and physically add or subtract from one another by moving the blocks to one side to see if 25 blocks remain. If students are having a really difficult time (lower level students who had trouble with 10), I could work one-on-one with them to help them count up or down the 100 chart with 10 again then possibly move to 20 then 25. But, having students do a similar task involving communication amongst one another during this follow-up task will allow for the students to become familiar with this format of teaching math and also be able to make discoveries about math on their own, creating an atmosphere where intrinsic motivation has room to develop. Also, having students talk with one another instead of the teacher doing the majority of the talking would allow for students to create a classroom culture just as Jill Bodner Lester had created within her classroom, read in the article “Establishing a Community of Mathematics Learners.”

Overall, this first experience with a mathematics lesson involving group discussion and sharing different student methods after the task has been worked on opened my eyes toward a new way of viewing mathematics and student thinking. Before teaching this lesson, all I knew and the students knew was the math format in which the teacher taught the method to use and the students would practice that method individually. Post-lesson and because of TE 401 discussions, mathematics in the classroom has taken on an entirely new role and atmosphere: one of open-endedness, encouraged variability, and student discussion.

**given sample student work in folder in TE 401 class.