Most successful learning goal
Based on scores alone, it would appear that the learning goal with which students were most successful was learning goal one because more students got these questions right than any other. However, learning goal five showed the most growth (69%) than any other learning goal. The success for learning goal one may just be that students were already familiar and just needed a refresher, where the results of the pre-assessment showed that no students knew how to complete every problem in the learning goal.
The one problem that some students got correct was the sum of the angles of a triangle. Only a few students got this right, but it would be explained by previous knowledge not by knowing how to complete the multi-step problem that gets the answer of 180.
One reason for this growth could be the group activity that we completed in class. Students worked in groups to use their knowledge of triangles to find the sum of the angles of other polygons. They took their polygon and drew a line from one vertex to every other vertex in the polygon to create triangles. They took note of how many triangles were made and the sum of the angles of the polygon based on the triangles. The students shared their results with the class and as a group we came up with a formula together to find the sum of the angles of any polygon. This activity and the practice done in homework solidified this concept in the students’ minds. Some students saw the same polygons so many times that they memorized the sum of the angles, but others knew the formula because it came from them and they were able to see and interact with the creation of the formula. Because this was a new concept, students who missed out on this activity likely did not internalize the process in the same way and were less successful on the post-assessment.
Another reason that this growth might have appeared was because of the alignment with the content of the unit to the learning goals in the unit. Looking at the unit as a whole, I was able to create learning goals that built upon each other. The knowledge gained about angles in the first lessons connected to the knowledge about interior angles of a triangle, which led directly to finding interior angles in other polygons. This progression of knowledge allowed students to find the answer sometimes even if they did not memorize the formula.
A2. Least Successful Learning Goal
In a similar way, I found learning goal 2 to be the least successful. It had the second lowest score overall and the least amount of growth from pre to post assessment with only 42% growth. It showed the highest scores on the pre-assessment.
One reason this was one of the least successful learning goals was the lack of basic skills in the class. I often found myself reviewing concepts when we were working on solving simple equations. Students would subtract instead of divide, multiply instead of add or add instead of subtract. A few students had little idea of how to use an equation let alone set one up, so we spent a lot of time reviewing these skills rather than working on the new skill. I feel like the review was needed, but may have taken away from instruction time spent on setting up the equations rather than just solving them.
Another reason this may have been a less successful venture was the instructional time spent on supplementary and complementary angles. I followed the lesson outline provided by the book even though it covered many topics in one lesson. I was leery about this, but forged ahead anyway. When delivering the lesson, the students were moving quickly with me and appeared to understand based on my observations of their practice. However, I believe presenting all of the information in one shot made it difficult for students to recall the information even if it was easy at the time.
If I were to teach this unit again, I would spend one day reviewing prerequisite skills. I would break up the first lesson into two lessons and make up for the extra time and the resulting increased need for differentiation with higher-order thinking activities and problems. I would also check up on homework more frequently and get in touch with parents more often to try to increase attendance and homework completion by those students who fell behind.
A typical schedule for my 1st hour pre-algebra class:
7:45-7:55 – Students work on warm-up problems and listen to announcements. Attendance is taken, homework turned in. The teacher circulates the class stamping correct answers on warm-up sheets.
7:55-8:00 – Review warm-up problem and answer questions from homework.
8:00 – 8:10 – Review any additional prerequisite skills if necessary
8:10 – 8:33 – Direct Instruction. Teacher models, asking questions as appropriate, students work as a class through examples and then practice similar problems on their own. Teacher circulates asking questions, checking for understanding, etc. This process is repeated for each new concept or building upon the previous concepts.
8:33-8:42 – Practice time – students start work on their homework, completing one problem from each section to give time for questions. Teacher circulates, answering questions, checking work and addressing prevalent misconceptions if they exist.
8:42-8:43 – Clean up if necessary
The structure of the class is set up to maximize instruction time. When students enter the class, they look at the board and see the objective for the day. They also see their homework assignment and write it down on the homework assignment sheet along with the date. They take out their warm-up sheets and begin working on a warm-up problem on the board. The announcements come at this time and I take attendance during the announcements to keep from interrupting class.
The warm-up problem can assess student ability and recall knowledge of a prerequisite skill or assess a concept learned earlier in the unit. Instruction begins as soon as students start working on the warm-up. Going through the warm-up as a class leads naturally into the discussion of the day’s topic. The rest of the time is reserved for instruction, which can consist of direct instruction, guided discovery learning activities or group learning. Most often the structure of teacher modeling, guided practice and individual practice takes place as it is often the most effective way to teach a mathematics class.
Depending on the lesson, I like to leave about ten minutes for students to begin their homework. That way I can correct any misconceptions before students practice the problems wrong at home. I ask the students to complete one problem from each section so they can ask questions if they need to. This usually gives me enough time to quickly review a concept if I find that many students are making similar mistakes. In other classes, we correct homework together, but in pre-algebra, I correct homework so that we have more time involved in instruction.
I have adjusted instructional time often in classes, especially in lower level classes. At times, I even adjust instructional time between different sections of the same class when one class takes less time to learn a topic than another.
One specific example occurred in my pre-algebra class. I was teaching surface area and volume of spheres. We had covered similar lessons on volume and surface area of other three-dimensional shapes with some success and a few students with computational problems.
I had prepared a visual presentation to help students understand surface area of spheres using an orange and its peel. I was also being observed by my clinical supervisor so I really wanted to follow my lesson plan exactly and show off my students’ progress through the unit. I went through the portion of the lesson on volume as planned. As I worked through the equations and sample problems the students answered everything correctly and seemed to have a grasp on the procedure. I gave them some problems to practice on their own and as I began to circulate the class, it became obvious that they were not connecting at all. Only one student was getting correct answers and only a few were following the procedures I had just finished teaching. I looked at the time and decided that I was not going to get to my prepared lesson on surface area and we would have to dedicate one full day to volume and one full day to surface area.
I stopped class and went through one of the problems, step by step starting with writing the equation, plugging in the radius and following the order of operations. After each step, I asked students to identify their mistakes. Nearly all of the students’ problems were with order of operations. All students properly set the equations up, but then just put them blindly into their calculators without paying attention to their calculations. We spent the majority of the remainder of class time reviewing order of operations and I emphasized the importance of writing out each step when solving such complex problems. I changed their homework assignment to just do all of the volume problems and since it was half the problems, I asked that students show every step no matter how small so that we could find any errors easily.
Based on my observations and the lesson, I made a warm-up that checked their understanding the following day and added an extra page to their chapter test review. This extra review had problems including rounding, order of operations and exponents. We reviewed the problems on their own and then within the context of finding volume and surface area. Similar skills were necessary for finding surface area so we were able to immediately use the skills that we reviewed.
This was the last topic in the unit so we were scheduled to have a test three days after the lesson. We had scheduled two days of review for the chapter and then the test the following day. The adjustment to instruction took away a day of review of chapter specific topics, but gained us instructional time to review knowledge that was necessary for success on the test.
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