- Why do you think we learn math in school, and why do you think math is an important or unimportant subject?
Jack French (teacher): Mathematics is a critical subject for everyone to know, therefore to learn in school. Mathematics permits intelligent interpretation of many topics one reads about, permits one to learn topics that utilize mathematics as a base...statistics, accounting, physics, many other aspects of science to name just a few, and the study of mathematics develops thinking skills, which are transferable to all other aspects of life.
Ian Bayer (student): I think we learn it in school for better understanding of it. So if we use a calculator, we know why it gives us that answer. It's important up to a certain extent, and after that it should be optional. Complex fraction isn’t something you would run into unless you choose that path.
Carol Funk (teacher): Math is used in everyday lives regardless of what we do for a living. Although we may not use the exact Math that is studied, in particular in the academic classes, we learn how to problem solve, how to apply our knowledge to gain new knowledge. Math explains how our world works. Math teaches organization of our thoughts and how to explain our thinking.
Brandon Jentsch (student): We learn math to better learn how and why things, live, exist, and just function the way they do. It is very important because it helps us better understand the world existing around us.
Gabriele Gonzales (teacher): Math is an integral part of life: you use it in doing business (buying and selling), in keeping time, in planning, and you use it in more academic subjects like chemistry and physics. Even Poetry uses numbers (meters).
Taking math in school also teaches problem solving skills and analytical/ logical thinking.
- For students: When learning a new material, do you tend to memorize the steps in solving a problem or do you try to understand the meaning and idea under each steps? and why do you do so?
- For teachers: When teaching a new material, do you just work through a problem and showing all the steps in solving a problem or do you explain or try to explain the idea in each of the steps involved in solving a problem? and why do you do so?
Jack French (teacher): If students understand why each step has been taken in solving a problem they are more likely to be able to solve similar problems, and eventually to solve many different types of problems, once their repertoire of procedures is large enough. They will not remember steps they have merely memorized.
Ian Bayer (student): The steps to get the answer. Uh... 'cause all I'm looking for is the answer. I'm not looking for understanding of the formula.
Carol Funk (teacher): The way we teach math has changed in the last couple of years. Rather than teaching to problem solve, we teach math through problem solving. Students solve problems using prior knowledge to learn new concepts. Multiple methods are emphasized and students communicate their methods to each other to understand that there is more than one method to solve any problem. Once students have shared how they approached a question I usually offer a formal summary of the concepts and clearly state what I expect to see when the students put their work on paper for future evaluation. I also stress the importance that students understand each step they use.
Note that today we are using manipulatives and models to explain new math concepts, something that was rarely used when you were in the junior grades.
Brandon Jentsch (student): I memorize the steps and why the steps are taken in solving a problem that way I better understand how it works so I can better memorize how to use that formula for solving the problem in the future.
Gabriele Gonzales (teacher): Explain ideas, the “why’s” so students can follow the thinking. This helps them understand and also teaches them logical thinking. Then summarize just going through the steps.
- What do you think of TPS (Think-Pair-Share) techniques where students break into groups of 2 to discuss the material before the class discusses it as a whole? How about groups of 5 students?
Jack French (teacher): Am not familiar with this procedure.
Ian Bayer (student): Yeah, I agree with the buddy system, sure. I don't know if I would want a group of 2 or 5, but it's better than a group of 20 students asking the teacher.
Carol Funk (teacher): We do this every day with the new curriculum and are trying to implement this into our senior classes. The size of the group depends on the difficulty of the task. Often we will start in pairs, then share with another pair, then share with the class. This gives students the opportunity to see alternative methods. It is amazing how creative students can be! By explaining the each other students improve on their ability to communicate using math terms and also strengthen their understanding of the concepts.
Brandon Jentsch (student): Groups of 2 would make sure everyone is doing something unlike groups of 5, but depending on class size 5 may be more appropriate, so my answer is 5 if you can make sure everyone is actively involved in working on the problem if not then my answer is 2.
Gabriele Gonzales (teacher): TPS make a student commit to an idea first, then takes away the possibility of public humiliation of being wrong by just comparing his answer to a partner. It allows him to defend his idea and forces both partners to think about their reasoning. It clarifies the concepts to both students before talking to the whole class. It would work with 5 students, too, if students can hold group discussions.
- Administering tests can be an effective tool in determining class proficiency before moving on to new material. How frequently do you believe tests should be administered?
Jack French (teacher): Teachers must receive fairly constant feedback from students in order to monitor the effectiveness of the classroom dynamics for both the teacher and individual students. If assignments are few, then tests should be many. If many assignments are issued, then tests can be less frequent.
Ian Bayer (student): That's a good question. Maybe at the end of every week or something. As often as the chapters move forward. I'm not a big homework guy. There is nobody at home to teach you, may as well be working at your own pace.
Carol Funk (teacher): Assessment is more than administering tests. A teacher must assess student understanding as they are first learning. This can be achieved by asking questions during the lesson, checking student work as they are working on the daily assignments, listening to students are they communicate their ideas to their peers…..
Students can assess how they are doing by checking their own work (assuming answers are provided). They can check with peers or the teacher to ensure they are on the right track.
I take in assignments daily to check progress and also give daily quizzes. Adjustments to lesson plans are made based on the results of the above. I also encourage students to let me know of any difficulties so I can deal with them before moving on.
By the time students get to the chapter test, they will have rehearsed several times on the assignments and quizzes.
Brandon Jentsch (student): Well probably weekly since stuff tends to be forgotten over weekends xp, but certainly after every chapter (basically after you finish 1 train of related thoughts and are about to move onto another is when you should test your class).
Gabriele Gonzales (teacher): Test at the end of the unit. Depending on the unit, intermittent quizzes are needed to check proficiency. However, they should not be the only means of assessment. Basically, they are mostly there to get grades.
- What are your views on traditional class lecturing versus a system where students could work at their own pace and each individual could spend extra time on material they found difficult?
Jack French (teacher): Individual learning is superior for the individual student in many cases. Students learn how to learn effectively and can operate at a pace that allows for effective learning. This school, for a time, had every student learning this way. No students were enrolled in non-academic courses.
Ian Bayer (student): I don't think there is a lot of teacher help at "work at your own pace". It's still 20 on 1. I think advanced students should take advanced math, and the rest should work as a class.
Carol Funk (teacher): In a perfect world it would be great to have students to work at their own pace and move on only when they have truly mastered prerequisite concepts. However we must deal with logistics such as large class sizes, time constraints (report cards, semestered classes…)
However with the use of models to help students develop better understanding, students who are not yet ready to move on to the use of algorithms can continue using models and manipulatives until they are ready, and in this way still be successful in problems.
Brandon Jentsch (student): Well while students being able to take their time and learn at their own pace is a nice idea, I have never had a teacher finish teaching us everything in a math book and I don’t think students have the time in the year to really learn at any slower pace than the teacher already has them going. If they need help they need to spend time after/before school with the teacher, get help from their parents, or hire a tutor.
Gabriele Gonzales (teacher): Difficult to implement in real life because most students who would take extra time on difficult material usually find MOST material difficult and would fall further and further behind. They would not complete the curriculum in time. Also may impede group work. However, some time and projects should be built in to allow for different speed of completion.
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