Kansas
The CASAS Competencies used in Kansas Adult Education programs include Communication(0.1, 0.2, 4.6), Effectiveness in Working With Other People (4.8), Demonstration of the ability to use thinking skills (7.2), and Demonstrate ability to use problem-solving skills (7.3)
Please share a story about how you saw or heard this core concept come alive for students during the Algebraic Thinking lessons.
As the student tackled the “in and out” tables, there was quite a bit of “crosstalk”.
It’s always so interesting to be in the room and hear their fears, concerns
and downright grumbling as though I was invisible. I believe this spontaneous
communication, however, did much to allay their insecurity of the algebraic
subject matter. The furtive glances among the students spoke volumes. When
one student began to formulate a plan for solving the table, the room took
on a “clannish” nature. Whispers could be heard throughout the room. Students
had been demonstrating a cohesive communication all along. As the news of a
pattern began to emerge, visible relief swept the room.
During on of our lessons in the first few chapters the students needed time to first process the language (vocabulary) of what is being taught. All being 2nd language learners their math backgrounds are as varied as the students sitting in the room. One class period the students were working on estimating answers, etc. The most serious student (also the best in math) suddenly announced that “teacher this is not a practical way to solve this problem.” He proceeds to the markerboard explaining the problem with the solution. Much discussion followed going from the solving the problem then reversing to the estimating—all students were satisfied and shared in the lesson.
Effectiveness in Working With Other People (4.8)
Please share a story about how you saw or heard this core concept
come alive for students during the Algebraic Thinking lessons.
The students didn’t appear to have questions about solving algebraic expression so I put several problems on the board and ask that each student solve the problems. After a period of time I found that the students were talking to each other asking questions and comparing answers. Those students who understand how to solve the problem were working with those who were having problems and explaining the process to the other students.
I also had a group of Hispanic students and I found that when they were having problems it was helpful to them to allow them to help each other using Spanish to communicate. It greatly improved their comprehension and understanding of how to work the problems as well as their fluency in transferring information to English.
During the lesson where we discovered what the value of pi is, the students worked together as they measured the circumference and diameter of circular objects. They marched around through out our 4 room center find the circular objects. Each student tried to find the most unique object.
It was then surprising to the students how the value of circumference divided by the diameter equaled approximately 3 in all cases.
The Algebra lessons encouraged group work. I have a small class. It is very easy for my students to ask each other for help or ideas on how to solve problems. As we would start a lesson, I would see the students looking at each other. I am sure they were thinking what is she going to make us do today.
During the Body at Work lesson over heart rate, I had a couple of students
who could not find their pulse. The other students encouraged them to look
in a different area or guided them on how to use their first two fingers. It
was great to see the students jumping in to help each other out.
These students were adults who get together twice a week for about six hours. They are not coming to class looking for a friend. Most of them have full-time jobs and families at home. It always amazes me how quickly the class comes together and forms a “family” of learners.
Students were working on the lesson with developing the rules for Lesson 2—Banquet Tables. I arranged the nine students into groups of three.
They worked together so well on the activities. I mixed math levels together and the upper level students assisted the lower level students. After I gave a beginning demonstration of the banquet tables the groups worked on their own with minimal prompting from me. I guess what amazed me the most was that the students really talked with each other. Often with group work, they fall into each one doing his/her own thing rather than interacting. With this lesson students were engaged—discussing, hashing our ideas. I saw one student patiently explaining an idea to another. Then enjoyed each other and the lesson.
One of my students came to my program at the age of 16. The first class session he made it clear to me that he was not there by choice and that he wasn’t going to learn anything. He was withdrawn and had difficulty focusing. His math was CASAS Level 3 (or C). He did sign up for the TIAN project and participated in both data analysis and algebra. For most of data analysis, he said he preferred working alone. However, by the time we began the last activity in our algebra sessions, he said he wanted to participate with the group. (This was the first time it was his idea to work with the group.) During this time period, he raised his CASAS level to 5 and passed all of his GED tests, thus receiving his referral to take the GED test.
It was a great thrill to witness his maturity, part of which included his ability to work effectively with other people. Because my classroom is geared to individual study, it was the TIAN project that provided the opportunity for him to develop this skill.
My students work together daily. They are very used to it simply because it’s almost a daily routine. One particular day stands out. I remember writing various math problems (adding and subtracting fractions with different denominators). Most of the students caught on quickly, but one didn’t. I continued with more examples and even wrote out each step in written form. For some reason, the student didn’t catch on. While continuing with more examples, the student became frustrated. The student who was sitting behind him, grabbed his paper and began teaching him. This brought in a new light for him. After she worked and discussed more examples, the student grasped on. After that, I put them in groups and allowed them to teach each other. They seem to catch on quickly this way, depending.
In my class, students ranged in ages from 16-55 and CASAS levels II-V. In the first activity that we did, I had seen the lower level students struggle with the multiplication part. As I watched, the students in the upper levels start to divide themselves up and work with those who were struggling. This group of students that I had seemed to work well with each other. They would give encouragement to those who wanted to give up and they sat right next to them reviewing multiplication tables with them. The upper level students did not want to move on with the lesson until whoever they were working with felt confident with what they were doing. Watching the students look out for each other meant a lot not just to myself but also to my students who felt less confident.
Demonstration of the Ability to Use Thinking Skills (7.2)
Please share a story about how you saw or heard this core concept come alive for students during the Algebraic Thinking lessons.
I taught lesson #1 “Guess My Rule” to a combined class of Level I and Level II math students. Level I students had no idea of the concepts of variable, coefficient, equation, etc. They had to adjust their level of thinking to learning an entire new set of definitions and terms and were intimidated by the process.
Level II students were more comfortable with the terminology, but had never been asked to come up with a “rule” or equation to match data on a table. They had to adjust their thinking from finding the answer to actually setting up the problem.
As I reflect on Lesson 9 “Phone Plans” I almost feel like I could write a little about the thinking, problem solving and communication skills my students used in this activity.
While students looked at description plans it was interesting to watch them think through linking a phone plan to a graph that visually represents the same information. Taking it to the next step, and having students then create an equation that reflects the phone plan began to make math real for the class. The students began at this point to work together and communicate what they saw and how they thought the information should look mathematically. Students processing skills were taken a step higher with this task. They had to think through presenting information in multiple ways.
“Demonstrate ability to use problem-solving skills” (7.3)
Please share a story about how you saw or heard this core concept come alive for students during the Algebraic Thinking lessons.
I believe problem-solving and thinking skills are closely related and are displayed simultaneously while working through the Algebra series. I am now in the middle of my second time through teaching “Guess My Rule” with groups of Pre-GED II learners (CASAS math scores 227-234). Both times when I began this lesson learners had a difficult time coming up with the In-Out tables. My first group had more experience with algebra, but my second group has had practically no experience with it.
With my first group, they were so used to simply solving equations they were given, they had a difficult time coming up with the equation given the chart. After going through a couple of examples of one-step equations, they were able to come up with the examples working individually and/or pairs. Then we came upon two-step equations. The learners became very frustrated and so did I. We stopped and went home. I studied the teachers guide and came up with a solution. We started the next day with group examples and the learners again took off with it. As a group the learners and myself used many different problem-solving skills, including working backwards using trial-and-error. We had a discussion about how these skills are helpful in solving non-math real world problems. Many times my learners said “I have never thought of this in this way.” This group would work together to come up with answers, while my group now, I have one learner simply giving another student the exact answer.
My second group has two learners who had Algebra in school. Two learners have taken my Pre-GED II class before. Four learners have had no experience with Algebra. We have had to spend much more time working on problem-solving skills as an entire group versus working in pairs and/or individually.
The students were understanding the in/out tables of the equations to find ‘X’ or ‘Y’.
On pg. 29 the students struggled with the problem
4 3 4 1 = 3
They had to come up with ‘which order of operation’ would go in between the numbers. There were 3 answers and they discussed it. They were:
(4 - 3) x 4 – 1=3
(4 + 3 - 4) 1=3
4 + 3 – 4 (1)=3
There was some discussion with students.