Using Research to Analyze, Inform, and Assess Changes in Instruction

The first part of the article that stood out for me was when she talked about how she “viewed instructional strategies as hit or miss” but how she had “come to realize that research can inform the selection and implementation of instructional strategies and, as a result, make a huge difference in whether a strategy hits or misses.” This stood out for me because I have heard many times this year that it is inevitable that some our lessons will flop. Taken the wrong way, one could easily just take a flopped lesson to mean that it just wasn't a good day and fail to reflect. I think that it is important to remember that while your lessons may not work out as well as you may have hoped, there is often something that we can adjust that can improve the likelihood of that lesson being effective. With our lessons, it is important to think about why something worked our didn't work so we can change our selection and implementation of instructional strategies.

What was also interesting was the section about higher-level thinking and how he implemented these higher-level questions into his class. In my experience, if you go through any math textbook or test, you will find very few questions requiring deeper thought. We did go through Bloom’s taxonomy in another course but, honestly, I don’t think I would have even considered it when writing a test or assignment just because it is something so uncommon in most math classes. This article really got me to think about how important it is to promote this critical thinking.

Short Practicum Experience with Math 8

On my short practicum, I taught a class of Math 8. It was a review session coving the multiplication and division of integers. It was interesting because they were required to learn how to use number lines and algebra tiles to show how to solve equations but this was something I had never learned. It was clear to me and their teacher going into the class that they were struggling with using these strategies. I began the class by asking naming certain areas of the chapter (ex. using number lines, using algebra tiles, solving multiplication/division problems using any method, order of operations, word problems, etc.) and how comfortable they were in them and they seemed to have a pretty good idea of what they needed to work on. I found it difficult to fit in everything that I wanted to do. I planned for a 60 minute review session but the quiz that my sponsor teacher planned took longer than expected. I did plan what I needed to cut out but I found that I still didn't have enough time for the shortened plan. In the end, I feel that it would have been better to focus entirely on one area that I had previously determined to be an area that they needed help on.

I also found that I should be more clear with my instructions - especially with grade 8s. It is becoming apparent to me that the more clear I am in explaining in exactly what I would like them to do, the easier it is to keep them on-task.

Mathematics Text

The first part of this text that I could relate with was the difficulty students find with the large number of mathematics symbols and graphics. As a student, I remember being able to develop an understanding of most material quite quickly for my high school years but when I started doing higher level mathematics and I began finding more and more weird symbols being used in the text and in class without explanation, I found myself confused and frustrated. I think that it is important that I remember this and do my best to help students gradually build these symbols into their repertoire. With all mathematical terminology, as this text argues, we should not be afraid to use it, but “allow students to grapple with their ideas and develop their own informal means of expressing them” while “avoid[ing] a premature rush to impose formal mathematical language.” I also liked the section about teachers contributing to students’ confusion of illustrations with definitions. I know I am guilty of always using the same picture for right angle triangles and I think it is a habit I should learn to break.

Something that I often think about is the use of textbooks. In my high school experience, most teachers used them - some a lot some a little - and one teacher was super anti-textbook (he is the Mr. I that I talk about in my memorable math teachers blog entry). I have been leaning towards the no textbook side but I think the text presents a valid argument that with proper instruction of how to use it, the textbook can be a valuable resource.

Microteaching Reflection

One suggestions for improvement that came up an many of the feedback forms was that it would be better to have clearer learning objectives. I feel that one thing that contributed to this unclarity was some confusion in how were going to handle the discussion of the logic games. We had discussed ideas as a group, then typed up a lesson plan, but when it came down to actually implementing it, we found that we lacked the time, we did a bit of improvisation. I found it interesting that a common positive on the feedback forms was that our lesson seemed organized which I’m not sure I would agree with.

I think our group did well on engaging our students. We did not find this too difficult because of our topic but it challenges us to find ways to make math fun for other topics. Another pro was that lesson was mainly student-led. I think that this is related to how many thought our lesson was engaging. I feel that if students feel like they are developing their own understanding and not just sitting listening to someone lecturing them, they are more likely to be engaged and learn. It is important, however, to make sure we provide adequate guidance in these more student-led lessons.

The Geoboard Triangle Quest

There are a few things I found interesting in this article. First, I found it neat how this question was posed by students rather than the teacher. I believe that it is important to foster an environment of curiosity and exploration.

Second, I thought that it was great that not only did the teacher foster an environment of curiosity and exploration but also one of collaboration. It is not the teacher telling them what should be done to find the answer, it is a group working with each other in both their mistakes and their successes. Through this, students learn from each other in a variety of ways and as the teacher noted, many more mathematical terms were used in the discussion which most likely would not have been thought about if students were working individually.

Third. Through this problem, I have seen more of how powerful student exploration in a problem like this can be. This problem allowed students to work with the concepts of congruency, reflection, rotation, right angles triangles, areas, distance and the list goes on. Not only that, but they were also able to learn to work as a group and communicate their ideas. In this case “less is more.”

American vs. Japanese Teaching Tendencies

I enjoyed reading about the differences in the teaching practices between America and Japan. The American structure is very similar to much of what I have experienced as a student. Review old stuff, introduce new stuff with examples followed by assigning homework: the standard format of one of my high school math class. I never have experienced schooling in any other country but I had the impression that mathematics education in east asian countries involved many drills which I am glad to hear is not exactly the case.

The main difference between standard American structure appears to be how teachers handle what Hewitt calls necessary information. The structure in Japan provides the evidence that students tend to learn better when they are given the opportunity to develop their own understanding rather than being simply given the steps required to solve a problem. A statistic that I found a little disconcerting is that U.S teachers gave twice as many definitions and procedures as Japanese teachers. I believe we should try our best to reduce the amount of stating processes which students have the ability to come to understand themselves. Instead, we should provide the activities and environment that enables students to establish their own methods of solving problems. If students struggle, its not a bad thing - it’s an important part of learning that both students and teachers must remember.

Arbitrary and Necessary

I found this to be a very interesting read. The distinction between the arbitrary and necessary was never really something I thought about before and being made aware of the realms into which these fall, I feel that I am better prepared to teach mathematics.

Being able to determine what is necessary allows me to direct to appropriate topics activities geared towards students using their awareness to understand.

In my experience, I was able to take what Hewitt calls received wisdom and using my awareness, transform this into necessary fact. I fully understand that this is not the case for all or even most students so I must be careful to give activities that open the students’ minds to the concepts and builds on their current awareness.

I must also note that if i choose to take an approach and accompany the received wisdom with an explanation, it is not productive to base my explanation on anything that falls into my awareness but not theirs. As Hewitt states, “A teachers’s explanation is often based upon the teacher’s awareness, and so may use things which students do not find evident - things which are not in the students’ awareness - and so the explanation will not be one which will help those students to educate their own awareness.” As I mentioned in an earlier post, “As I work to become a teacher, I must not forget how a student thinks. Put myself in their shoes and remember what it is like to not know.”

A question I have is whether informing students of necessary mathematics content (received wisdom) can be a preferred method over the alternative in which the teacher works within the realm of awareness.

The Locker Problem

I first solved a problem similar to this a couple years ago. The problem I was given involved 100 lightbulbs and I initially just went lightbulb by lightbulb and seeing whether each was on or off. I remember being too lazy to get out a piece of paper so I just did it in my head. By the time I got to around locker 20, I saw a pattern - all lightbulbs in the positions of perfect squares were on. I realized that this was because the number of times a lightbulb was switched on or off was equal to the number of factors it had and only perfect squares had an odd number of factors (all factors of a number have a corresponding factor which when multiplied by each other equal the number except the square root of a perfect square).

I realize that a student attempting my method may not see the pattern as quickly as I did and may eventually get frustrated and give up. I also imagine that other methods might be more popular. I have seen many listing out the lockers and instead of going locker by locker, they went student by student changing the status of each locker as each student went by it. If a student had this method in mind, the sheer number of lockers might be daunting. If i approached a student who told me that they were having trouble with the number of lockers, I would suggest that they focus on maybe the first 30 lockers and see if they notice anything. Completing this task would hopefully allow them to see a pattern and then hopefully give them the opportunity to think about why that pattern is the answer.

As an extension, you could specify various lockers that you wanted open and then ask for which of the 1000 numbered students should be dispatched to open/close their respective lockers.

Difficulties and Challenges Involved in the Chess Board Problem

As a student, I solved the problem first by counting individual squares and then realizing  that there was a pattern. Using the pattern, I was able to solve the problem relatively quickly.

As a teacher, I can see students finding difficulty when trying to determine the number of squares larger than the 1x1 squares. Counting these becomes difficult past this stage as the squares begin to overlap. I would encourage an approach involving finding patterns. To guide them, I could start by helping them with the first step in determining a pattern.

To modify this problem to make it more difficult, you could add another dimension and ask for the number of cubes in a 8x8x8 cube. or ask for the number of triangles in a similarly structured triangle where patterns would be more difficult to find.

Memorable Math Teachers

One of my most memorable math teachers was Mr. P who taught me in grade 11. What I remember most about him were his stories that he brought to math class each day. Some were about his son, some were about his life before coming to Canada and some were about totally random topics. Very few of these stories were related to math but I enjoyed going to math because I knew that I had a teacher who knew how to make us laugh. I must say that his teaching style was not the best. He took a more traditional approach where he would give us a lecture which was followed by class time to work on textbook problems but I came out of it fine because I was able to develop an understanding of the math quite easily.

The following year, I had Mr. I for calculus. He taught us in a way that I had never experienced before. Classes were very interactive and there were times where we were even be using our bodies and voices simultaneously to learn. Lectures, I would say, were non-existent. Even the formal testing I was used to was not present. Our understanding was assessed mainly on our class participation and a few very informal quizzes. I found this teaching style quite interesting and I would say that he succeeded in getting me engaged in the subject matter. I have observed that the lack of formal assessment was a big problem for some students and even some teachers. Students would often question why they got the mark that they got. Furthermore, some students who were very engaged and given high marks were given permission to take courses designed for students showing proficiency in math. There were some cases where students were not preforming well as their math skills were not at the A level which Mr. I had evaluated them to be at. Nevertheless, I believe Mr. I was able to engage his students and encourage us to think in new ways and I will remember him for that.

Experiment and Learn

In my highschool, like the many described by Gerofsky, we had only two teachers for a school of 1500 fully qualified to teach mathematics. I was brought up in more conservative system of mathematics education. Fortunately, this worked well for me. I can easy understand with what was said about those who have “found ways to make sense and understand the mathematics they were presented, and expect that anyone who is good at math should be able to succeed as they did, under a similar system.” Does this mean that how I should teach should model the education practices under which I was taught? As many would argue - no. I must remember there exists a diverse range of learning styles and that a conservative mathematics education will very likely make it difficult or even impossible for many students to thrive. Can I expect students to extract an relational understanding of the material from a class involving mainly drills and memorization of facts and formulas? I don’t think I can.

The alternative to this would then be a progressive approach - a stance involving experimentation which can be “messy, uncertain, and unsettling.” I can also understand the parents’ “worries that their children were being shortchanged by teachers experimenting with their education.” I believe this is a necessary risk if we want to provide students with the best possible education. We must set the example that it is ok to take risks, mess up and learn from our mistakes. It is very important that we learn. We have the responsibility to educate and if each educational experiment we conduct fails, I believe we are failing even if we establish the belief that mistakes are ok.

In short, I believe that, as math educators, we must not be lazy and conform to the traditional ways in which we have been taught. Instead, we must make the effort to challenge students in a variety of ways while at the same time learning and reflecting on our efforts.

Communicating Understanding

What I got most from Thurston’s article is encapsulated in the following line: “The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.” I could easily identify with the image Thurston describes of an audience getting lost in a colloquium talk within the first 5 minutes and sitting silently through the remaining 55 minutes. I can see how effective communication of mathematics with students has does not depend on how big your words are or how vast your vocabulary is but depends solely on how you are able to connect to students in a way that they can understand. We must be able to see the diversity in our schools: diversity in ability and diversity our ways of learning. Considering this, we should see that the language we use should be understandable by students, not only by you. Furthermore, our teaching must accommodate all learners - visual, auditory or kinesthetic. I enjoyed the example of the various ways that the derivative could be understood and how spending time connecting these ideas promoted a fuller understanding of the topic. 

I must keep this in mind: As I work to become a teacher, I must not forget how a student thinks. Put myself in their shoes and remember what it is like to not know.

Benny’s Rules: The Problem with IPI Mathematics

From Benny’s Conception of Rules and Answers in IPI Mathematics, I found Benny to have a thorough frustration stemming from his view of what mathematics was. What I found most illustrative of this is when we are told that he said, “I am going to look up fractions, and I am going to find out who did the rules, and how they are kept.” The rules and their inflexibility in his view of mathematics is certainly not the stance I would my students to adopt. Perhaps what is lacking the most in the method of instruction referred to in this article is the discussion with peers and teachers. I believe without this, the study of math can very easily become what I had become for Benny: a fixed set of rules applied to a variety of problems to arrive at the right answer. It seems to me that the effective teacher-student interaction is key to developing an attitude of relational understanding which I believe the joy in math comes from.

Relational Understanding and Instrumental Understanding

What struck me when reading Richard Skemp’s Relational Understanding and Instrumental Understanding was how easily I could relate to the two meanings of understanding. From my experience helping others in math, I found that a big difficulty for many was recalling what they had learned in previous grades or courses. It appears that they do not have the relational understanding explored in the article but only once had some instrumental understanding that has since disappeared. It is clear to me that their background of instrumental mathematics has led them to approach pretty much all mathematics problems with a similar way of thinking however, at higher levels of math, this becomes very difficult as to apply these new rules, they must remember some old rules which have fallen out of their memory. I feel like math to them seems like nothing more than memorizing a bunch of formulas and rules and to break this way of thinking is not an easy task. I’ve also felt at times a need to go back and help them relationally understand the basics but this proves to be a very, very time consuming task and I find it quite understandable why many who “miss a step” have trouble catching up. I think it is a very important task to have students understand relationally as much as possible starting from beginning because from my experience, memorizing is terribly unstimulating while having a true relational understanding can be fun and lead to further exploration.