Reflection on “Teaching the Marked Case”

Dave Hewitt introduced a rather unusual method for teaching math. He used pattern recognizing to present a material from a really simple basic to a more complex idea. Also, the idea of a variable ‘x’ was brought up unconsciously. I found it interesting that, even without an explanation of what a variable is, the students could grasp the idea on their own. Also, the lecture style was really interactive. It is hard to not pay attention to the teacher in the class.

I like how he let the students “self-check” their own answers. It boosts the students’ confidence and thus more likely to be interested in math and be more engaged with math problems. In addition, the students were given the chances of being rather creative in making up different ideas on how to “arrive” at the same value. However, I found this lecture style too slow paced. Some students may be bored during the lecture.

To conclude, I think this lecture style may work, but not in all kinds of topics. For each different topic, we need to think about which teaching style would fit in with that particular lecture, as no lecture style fits all the needs in the topic.

Reflection on Battleground Schools

Since 1900, mathematics education has been viewed in two different ways: progressive and conservative views. The progressive view focuses on the understanding of math, while the conservative view focuses on the fluency of math. Repeated arguments and conflicts on these two different stances have taken forms in the twentieth-century for many times.

There have been three reforms in mathematics during the twentieth-century. Progressivist reform from 1910-1940, led by John Dewey, focused on doing mathematics in contrast with the old “pure memorization”. Dewey’s techniques involve stimulating materials that motivate the students to inquire about math.

The second reform, the New Math (1960), focused on improving mathematics education from K to 12 levels. This reform pushed all the topics from university mathematics on to the K – 12 systems in order to prepare the students with the basis to become scientists.

The third reform, Math Wars (1990s – Present), is based on the NCTM Standards. Important features of the reforms include: “the development of curricular materials, professional development workshops, and courses for teachers, new modes of assessment, and the involvement of parents and educational technology initiatives”. (pg. 399) During mid-1990s, the Conservatives started to question about the Progressives’ way of teaching mathematics. Since then, there have been constant battles between the two different stances.

It was interesting to read about math phobic. It is rather depressing when the general public has such negative presumptions about math. Also, it is disheartening about how the majority of elementary school teacher lacks understanding of math concepts. If the children weren’t motivated and encouraged about math at the elementary level, it would be hard to get the children interested in math again and lift their fear about math at the secondary level.

Interestingly, my other education professor today quoted a rather intriguing quote to think about:

“The lecture is the process by which the notes of the lecturer are transferred to the notes of the student without passing through the mind of either.”

MAED 314A Interview Reflection

When I was in high school, I hardly work with other students during math classes. We just sat and listened to the teachers talk. There was no class involvement. Thus, it was interesting to see that both students we interviewed support the idea of incorporating TPS (Think-Pair-Share) techniques in their math classes. I always thought that TPS only works with classes like literature where ideas are diverse. Apparently, the students do prefer to have discussion on anything they learn, whether literature or math. TPS techniques let the students engage in active thinking environment which can help the students in understanding the materials. These techniques are also beneficial even in math classes as our interviewed students suggested. Thus, TPS is a technique I would definitely try to incorporate in my future math classes.

During the group presentation on Friday, I noticed how every group asked approximately the same questions. Thus, these questions had similar answers:

“A good teacher is one who has sense of humor and good organization.”
“As a teacher, you need to be well-prepared and need to have good classroom management.”
“Stay positive.”
“Don’t be monotone!”
“Relate math to real life to get the students interested in math.”

The answers are so stereotypically ideal. There wasn’t much “news” or “ideas” regards to teaching. In my entire education courses, I have been told several times of the same idea: the teachers need to be organized, prepared, thoughtful, caring, devoting, motivating and many more...

One thing I liked during the group presentation is the example of collecting homework based on the results of quizzes. I like this idea because even if a student failed a quiz, by proving that they did or at least tried the homework, it demonstrates that this student is indeed trying to understand math, whether relationally or instrumentally. As compared to the students who did not pass the quiz and did not do the homework, this one student should at least be applauded of his or her efforts in math.

MAED 314A Interview Summary

• Why do you think we learn math in school, and why do you think math is an important or unimportant subject?

The general consensus is that Math is important to understand the world around us, and is a critical prerequisite to learning essential skills such as statistics, accounting, physics, poetry, etc. Additionally Math helps to develop critical thinking and problem solving skills.

• 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?

Our students, who both struggled with Math, had differing opinions on Instrumental versus Relational understanding. The first student, who learned using a "work at your own pace" system, believed that the best method for him was 100% instrumental learning, since he viewed mathematics as an unnecessary learning exercise, and had no interest beyond passing the course. Our second student believed that if he had a Relational understanding, that it would be easier for him to develop (or "memorize") the Instrumental understanding.

• 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?

Two of our teachers believe strongly that Relational teaching leads to a stronger understanding of the material than Instrumental teaching, which in turn allows the students to better apply their knowledge to a wider variety of problems. Our third teacher believes it is important to emphasize a variety of teaching techniques, both Relational and Instrumental, combined with classroom discussion and a final summary of the Relational concepts.

• 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?

Two of our three teachers were willing to comment on TPS techniques, but for different reasons. Our first teacher believes that TPS can help both as a classroom management tool, as well as help reduce the possibility of public humiliation that occurs when a student answers a question wrong. Our second teacher supports TPS techniques because the students take a more active role in their learning. She finds it inspires creativity in the students, improves their ability to communicate using math terms, and strengthens their understanding of the concepts.
Both students believe TPS techniques would help with classroom management. One student expressed concern regarding groups of 5, where not everyone may be actively involved.

• 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?

Our teachers as a whole believed that constant assessment is necessary, but did not believe that tests were essential to administer frequently. They all stressed that alternate methods of assessment were just as effective.

Both students explicitly stated that tests should be administered at the end of every unit, and suggested one test a week as a good pace. One of the students specified that he prefers tests to homework assignments as a means of assessment.

• 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?

Our teachers agreed that a "work at your own pace" system would be nice, but is very difficult to implement due to time constraints and class management. Although not included in the written response, the one teacher, Mr. Jack French, did mention in a phone conversation that the administration and parents pushed heavily against this system due to students having to take a Grade 12 math block to finish their required Grade 11 mathematics, which by the program requirement, they eventually passed with 80% or better.

One student expressed concern that a "work at your own pace" system would result in the students getting behind, while the other student was concerned about teacher unavailability.

MAED 314A Interview Responses

• 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.

5 Math Burning Questions

1. Why do you think we learn math in school, and why do you think math is an important or unimportant subject?
2. 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?
3. 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?
4. 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?
5. 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?

Funny Math

Some math related jokes found on the internet:

Reflection on "Using Research to Analyze, Inform, and Assess Changes in Instruction"

The article was fascinating as we see how other math teachers self-assess and modify their teach styles when possible. This shows how teaching is an ongoing process; it is always changing and adapting to the new curriculum and new situation.

I also find the article interesting when we get to see the comparison of the old test to the revised one. The wording of the problems can impact a lot on how we approach the problems. By having questions address to the different levels of Bloom’s Taxonomy, the students can be so much more engaging with the materials. The students would really need to understand the material if they want to do ‘well’ in math. This then can trigger some students’ initiatives.

As well, Robinson pointed out how Jigsaw group helps her students in learning about math. Having discussion groups during class, Robinson’s students become more involved in the whole process of math. When the students become more involved, they become more motivated and thus have better understanding of math. The students will be more willing to learn math on their own.

However, Robinson’s job is far from done and same goes for us. Teaching is ongoing. There is no perfect teaching as no two groups of students are the same. We need to constantly self-assess and put ourselves in the students’ shoes. Concern about the students!

Two of My Most Memorable Math Teachers

• My math 10 teacher simply teaches directly from the textbook. It was so boring. Not only that, he often make mistakes doing calculations even when he copies directly from the textbook. He could not figure out how he made the mistakes. Yet everyone in the class ‘liked’ the teacher because we were always able to get some free bonus marks due to his mistakes on test questions.

• My math 8/9 teacher always makes corny jokes during the class. Math class suddenly became a comedy class instead. I can hardly learn anything from him because he is always busy making jokes and trying to make the class enjoyable.

Both teachers were bad examples of math teachers. They try too hard on improving the class in the wrong way. The focus wasn’t on math anymore but something else. These two math teachers are memorable because of their strong characteristics; one being funny and one being super careless.

Microteaching Evaluations

Peer Evaluations

• Learning/Teaching objective: Clear Objective
• Bridge/Intro: Product was shown
  
• Pretest of prior knowledge: Asking questions; good
• Participatory activity: Involvement was fun
• Post-test/check-in on learning: Helping person with trouble; wonderful
• Summary/conclusion: Handouts given was good; can enjoy in future by making own stars
• Strengths of this lesson: Organized well; hands on are good; handouts given leave strong impression of the lesson
  
• Areas need further work and development: Trouble with students left and right with my left and right; slow down at tricky parts and a bit hard to follow

Self-Evaluation

• Things went well: Successfully getting students’ attention and getting them interested; organized; took apart the product to show how it is made, so that students have rough idea about how are they going to make the same origami
  
• Improvement: whether to wait for all students to settle down before starting, or not; left or right? My right vs. your left; sit closer together so the students can see the steps in details more clearly

Comments: From my groups today, I learned that visual aids help a lot in explaining the concept or the ideas. It was a bit hard to follow from the teachers who just keep on talking. It is also important to catch the students’ attention at the beginning of the lesson. I also noted how asking questions help the students get into the topic and think about it, rather than just listening about the topic.

Ninja Star Lesson Plan

1. BRIDGE: To get students interested by telling a story about ninjas:

   “Imagine you are a ninja-in-training. What will you do if you run out of ninja stars when you are on a mission? All you have is a stack of origami paper! What should you do?”

2. Teaching OBJECTIVES: To get students interested in origami
3. Learning OBJECTIVES: To fold a ninja star with a single piece of origami paper
4. PRETEST: To ask the students,

   “Is anyone into origami? What can you make?”

5. PARTICIPATORY Activity Ideas: To show and let students follow step-by-step on how to make a ninja star
6. POST-TEST: To let students fold a ninja star on their own with a handout of instructions for reminder
7. SUMMARY: To show them other types of ninja star and other types of origami to get students interested in making more origami; to let students ask questions about what they learned
8. PREPARATION: To bring origami paper; to print instruction handouts from here
   

Reflection on Relational Understanding and Instrumental Understanding

After reading the Skemp’s article, I realize how I have always been taught instrumentally. However, I never felt the instrumental understanding was forced on me, as I automatically tried to understand the material relationally. Thus, prior to hearing about the two different meanings to the word ‘understanding’, I have always regarded ‘understanding’ as identifying why a method works and realizing how it works in practice. I do not see how instrumental understanding is a form of ‘understanding’. To me, instrumental understanding is just pure memorization and an imitation of relational understanding.

Upon reading Skemp’s article about relational understanding and instrumental understanding, I realize the difficulties in teaching relational understanding to high school students. Most high school students are more likely to tend to instrumental understanding as it is easier and faster to achieve good grades on exam. I recall my friend telling me, “Who cares? I got it correct and that is what matters.” This gives rise to the question, “how do we make students learn relationally?” Maybe we can try to lay the basic foundation with instrumental understanding and build on it with relational understanding. Maybe we can try relational understanding right off the bat. Maybe we should just teach instrumentally if that is what the students need. Maybe how we teach the material is dependent on the topic we are teaching. I guess the answer will come as we gain more experiences as teachers.

Lastly, I like how Skemp use the route-finding example to describe the two types of understanding. This example clearly demonstrates how instrumental understanding is useful for quick information, though unhelpful if one get lost in the middle of a step.

Interesting quotes from the article:

“[Instrumental understanding] is what I have in the past described as ‘rules without reasons’” (pg. 2)

This quote describes directly about what instrumental understanding is.

“If we were now to say to him “You may think you understand, but you don’t really,” he would not agree. “Of course I do. Look; I’ve got all these answers right.”” (pg. 2)

This quote shows how instrumental understanding gives a false delusion to students in making them to think they understand. In reality, they are just repeating the memorized instructions. If we give the same student the same question with alteration in the wording, he may not have been able to get the answers correctly. The rigorous step-by-step methods will fall apart in this type of situation.

“All they want is some kind of rule for getting the answer. As soon as this is reached, they latch on to it and ignore the rest.” (pg. 4)

Some students fail to notice the fact that the process of getting the answer is just as important as the answer itself. Those students will be thinking, “If something is not broken, why fix it?” or “If I can get it right this way, why learn anything else?”

““[Relational mathematics]” is easier to remember.” (pg. 9)

Indeed. I used to have trouble remembering what formula is used for calculating the surface area of a sphere. However, after learning calculus and realizing calculating integral gives us “the area under the curve”, I can now use this same idea to surface area of a sphere. By thinking about how surface area is like a “curve” and volume is like the “area”, I can derive the surface area directly from volume without having to memorizing another set of formulas.

“Difficulty of assessment of whether a person understands relationally or instrumentally.” (pg. 12)

I agree with this quote. When a student answers a question, we can only infer to what the student is thinking by what was written down. Whether the students understand relationally or instrumentally, we won’t know unless we ask each of the students directly about how they come up with a solution. Students with instrumental understanding could have answer correctly because they were simply following the steps. Students with relational understanding could have answer correctly because they truly understand how to work out a problem. At the end, we won’t know unless we talk to the students individually which can be time-consuming.