Math Methods Blog

Prompt: Throughout the entire book, change is a driving force.  Whether it is talking about change in voting rights or the way mathematics it taught, it is constantly talked about.  How can you apply this theme of change to your future mathematics classroom, not just with the Algebra Project?

I think that a math journal can be very effective in the classroom.  This journal that I have been doing throughout the semester has made me think about all the readings I have done, and really reflect on the prompts that I was given.  If students reflect on the class period every time they learn something new, it will give me an idea about the misconceptions they may have formed, as well as provide me with an idea of their level of understanding.  However, I need to make sure that these blogs are not a lot of extra work.  I think a quick three to five sentences is manageable for students, and since the prompt will almost always be the same, it will be like second nature for students to answer the prompt.  I like the idea of students reflecting on the class because they will have to reflect on their lives throughout the rest of their schooling and working careers. 

The main problem I think I am going to have to work with is what students use as their math journal.  Ideally, I would like it to be some sort of blog because then those students who are not verbal learners can still express their thoughts through pictures and words.  There are two main concerns with blogs though; 1.) are they are safe type of blog that I can easily monitor so students do not get in trouble? and 2.) do students have computer access at home in order to complete the entry?  I know that there are some blog sites out there that are perfect for students because they can have super strict privacy settings and be safe for students to use.  However, not all students have a computer at home with internet access to use a blog site.  Students can always write in an actual journal, but it is not as fun as a personalized blog that can easily use pictures and sound.  There are ways to work around the no computer at home problem and I would simply have to consider those if not all my students have internet access.  The main thing I can do is provide time for the journal entries to be done in class.  This way it is not extra work for the students and I know for sure that everyone is writing an entry.

There are a few differences when it comes to teaching middle school as compared to high school.  The first major one is the age of the students.  This may be a rather obvious difference, but I feel like it is over looked.  In high school we are preparing students for college/working life, while in middle school we are teaching students more basics and preparing them for the college/working concepts they will be getting in high school.  Not only that, but we are teaching middle school students valuable social and developmental skills that we tend to stray away from in high school.  Skills like team work, listening to others, good note taking, etc. are focal points in middle school, but get pushed to the background in high school.

I have not had that much experience when it comes to gifted and talented students in math classes.  In high school, I'm not even sure if we had a gifted and talented program. If we did it was a secret society because I knew nothing about it.  I wish we had one though.  I would have thoroughly enjoyed being challenged in math because I simply sat in class bored out of my mind and easily did the work.  A gifted and talented program would have definitely made my mathematical experience in high school more beneficial.

Appendix 2 relates exactly to the CCSS because it is a shortened version of the ideas and concepts that students will be learning in those grades.  It includes algebra, geometry and spatial reasoning, measurement, number sense and quantitative reasoning, probability, and proportional reasoning which all are a part of the CCSS to some extent. Other than that, I do not see how it applies to the CCSS too much.

It does kind of relate to my unit, but in a very basic sense.  The ideas mentioned in the algebra section of the appendix are skills that the students supposedly should have learned in junior high.  I am going to have to use these skills and concepts as check points at the very beginning of the school year to see where there are any misconceptions and lack of understanding with prior knowledge, as well as make sure students can do the prerequisite skills that the appendix says they should know how to do by the ninth grade.  Other than that, this appendix is just a great thing to look at the see some of the prerequisite skills that will be needed in my lessons and also what the students "know" before entering my class.

There are five different teacher activities mentioned in Appendix 1, but I think that three of them apply the most to my unit:

1. Teacher encourages reflective discussions that elicit possible misconceptions.

I intend to have students discussing with each other as well as myself throughout the entire unit.  In lesson one they are asked to talk about what they observed and did during the balance activity and how they think it relates to solving equations.  I can hopefully weed out the preconceptions about equations with the very first activity of the unit.  The discussion might not elicit explicit misconceptions, but I will be able to pick them out and correct them early on, since the concept that students are meant to get out of lesson 1 is a huge part of solving equations and inequalities.

3. Teacher creates a nonthreatening environment for the analysis of students' errors.

N ow I do not have anything specific in my unit that addresses this teacher activity, but I plan to always have a safe environment in my class.  I want students to feel comfortable to present their ideas to the class and not worry about being ridiculed because they mess up or have a misunderstanding about something.  I plan on doing this by showing students that it is alright to not know the answer to something or not arrive at the correct solution.  It is obviously not the end of the world, and it students feel like they will not be ridiculed or graded poorly because of their ideas, then I believe I will have achieved the nonthreatening environment.

5. Students stop to look for hints, specific key words, and teacher's support for the application of new concepts.

I want my students to come to me for help and assistance.  How else am I supposed to teach them the most that they can learn?  Sometimes students, like myself, just need one little hint or key word to have the whole problem come together for them.  I will willingly provided students with the necessary assistance that they need to solve any problem.  And if I am expecting them to apply a new concept, then this aid is a necessity.  Without it, students may do wrong work or follow misconceptions that will only hinder their knowledge not help them.

This chapter was all about assessment, mostly the formative kind.  A lot of the strategies they mentioned I have heard before and have used in my lesson in some capacity or another; all except the student interview one.  This was a very interesting idea to me.  I have always heard of student surveys or questionnaires to get specific answers from students, but a student interview seems to complicated to actually use.  I mean, actually sitting down and having a conversation with every single student about a topic does not seem plausible.  There is not enough time in the school year to perform this type of assessment.  Also, what would the rest of the class do while I am interviewing one student?  It does not seem practical.  I love the idea of actually getting students' feedback right from their own mouths, but I do not feel like students would give their honest opinions, ideas, or processes because they will fear a lower grade for the wrong answer.  I would have never told a teacher exactly what I thought about an assignment, or my problem solving process in case I looked ridiculous to them.  I can make the classroom as safe and as comfortable atmosphere as possible and students will probably not talk openly with me.  I like the concept of the interview, but I do not think it would work in my classroom.  And I do not think I would actually use.  I might use the idea of asking students questions, but do it autonomously so that the students are not as pressured.  There are many other ways to check students' understanding and problem solving process without putting them in the spotlight of an interview.

Dear Parent(s)/Guardian(s),

    Your son or daughter is currently enrolled in my mathematics class at _______ High School for the 20__ - 20__ school year.  Your student will be expected to do a variety of mathematical computations and explore different mathematical concepts and ideas.  Students will also be expected to use a graphing calculator to investigate some of these concepts.  I know there is a controversy over the use of this technological tool; however, this letter is meant to help displace some of that fear you may have.

    I have done a little bit of research in this area, so I can provide you with some facts.  It has been rumored that graphing calculators simply provide students with a way to cheat their actual mathematical knowledge and learning.  However, according to S.S. Choi-Koh, there is no evidence to support this undermining of math learning.  Actually, in the study that Choi-Koh did on his one student, the use of a graphing calculator enhanced the student's knowledge.  This same enhancement can be true for your student as well.

    Graphing calculators turn an everyday math classroom into a lab, like in a science class.  This technology enables students to explore a concept by looking at a manipulating graphs and tables.  In the case that Choi-Koh observed, the student got to literally see the difference between the graph of y=sin(x) and y=2sin(x).  This student got to physically see the difference between the graph and table and see what the coefficient does to the equation.  This instant visualization enables students to make inquiries and conjectures, and then be able to test them.  This type of learning is much more student based, instead of me simply standing up in front of the class and lecturing them on the differences.  This type of exploratory learning also appeals to the student.  Imagine your son or daughter actually enjoying mathematics. 

    Not only does graphing calculators provide multiple representations, like I mentioned above, but it also is a great way for students to check their work.  Choi-Koh's case study learned how to graph different trigonometric equations by seeing what each part of the equation did to the original on the calculator.  Then, after he learned that, he could graph the equations by hand much more easily.  Then the case study student was able to check his work.  This is probably where a lot of the negative stereotypes for calculators come from.  If students have the ability to check their answers, then they must have the ability to simply get the answers.  However, if the class is taught correctly, the calculator will help teach the students something they can do easily on their own.  Checking their work is no different than spell check on a word processor. 

    Graphing calculators are going to be the new norm in every single mathematics classroom soon, and I believe they will greatly benefit your son or daughter's ability to learn math more effectively.  It even has the potential to get students interested in math again because they will be learning through exploring, instead of just listening to me. With proper questioning, students will be able to test their intuitions about a topic and then confirm or contradict their conjectures.  This is what math is all about; exploration and inquiry.  Graphing calculators are the technological tool to help transform the math curriculum for the better.

    If you have an other questions or concerns about graphing calculators or your student's interaction with them, please feel free to contact me. 

    Sincerely,

    Ms. Mykayla Stoutamyer

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Unfortunately as a math teacher, I am at an automatic disadvantage when it comes to motivating students.  Students come into a math class already hating it, so it takes that much more planning and effort on my part to grab my students' attention.  The Art of Motivating Students for Mathematics Instruction, by Alfred S. Posamentier and Stephen Krulik, provided me with great general ideas and also great specific examples as to how I can motivate my students to learn in my class.  There are a variety I can use in my classroom, but I am going to focus on the general nine motivational strategies and the few that appeal to me the most for my lesson.

Presenting a challenge, enticing with a "gee-whiz" mathematical result, and indicating a void in students' knowledge I think are the most beneficial to a math classroom as they tie together very well.  Students tend to think that they know everything about a topic just because they have seen something similar before.  However, this is obviously not the case because no one knows everything.  If I was to indicate a void in students' knowledge, it would intrigue them to learn more.  Algebra, especially the equations and inequalities piece, always has something new to learn.  By showing students that there is more for them to learn, it will maintain their interest in the subject.  I think presenting a challenge to them ties in nicely with this because most of the time the challenge leads to an interesting math fact or idea that is interesting.  And besides, who doesn't want to solve a difficult problem and say they did it?  There are a variety of interesting and challenging algebra ideas that students can figure out and can then trick their friends and family with, which ties in directly to introducing a "gee-whiz" idea.  These three techniques work well with one another and I plan on using them in a combination with one another.

The other very important motivational technique that is mentioned in this book is finding patterns.  Mathematics is full of patterns and when students discover these on their own it is rewarding to them and also enables them to retain the information better.  In my very first lesson students are going to find the patterns of solving equations using balances.  Students will hopefully be able to see these patterns, which relate directly to being able to solve equations algebraically.

Problem solving is huge in mathematics, but I didn't even start learning how to problem solve until I reached college.  I was always given the necessary information needed to solve a word problem, which is definitely not the same as problem solving.  I think the best way to use problem solving in any lesson is to incorporate it either right at the beginning of the lesson when students have no idea what skills they will have to use to solve the problem, or part way through a lesson where students have some idea what skills are going to be used, but they are going to have to use their imagination and outside skills to completely solve the problem. 

If I was to use the Problem Solving Method part way through the lesson, it enables me to equip students with the skills of solving equations and inequalities which is kind of necessary in actually solving some real world problems that deal with these foreign objects.  Most students do not even know what an equation does let alone how to solve them, so if I introduce those ideas to students, then they can play with and imagine different ways to solve a problem.

However, I could also simply start a lesson with a problem and see how students approach it.  Then, when they come up with these different and unique ways for solving, I could show them how their process actually uses the concept in the lesson or could be made easier with the concept.  This would allow students to actually problem solve, but would also introduce the lesson.  Using problem solving this way also enables me to use more complex problems later on that involve not just the ability to solve equations and inequalities, but previous math skills as well.  Because students have already solved problems without the knowledge of the lesson and were taught how to solve it easier, the more complex problems would allow students to actually see the easier way, as well as make them think about a solution for a different type of problem.

I think both of these ways could be used in my unit, depending on the students and the way that I think would interest them the most.  I could also use both methods.  The most important thing about using the Problem Solving Method, is to actually have the students think about what they are going to have to use and do in order to solve the problem.  Both of the ways above enable students to think on their own.

One of the great aspects of teaching is when a student (or students) realize their own misunderstanding(s) and then they start to understand a concept.  It is really rewarding as a teacher to witness this classroom miracle.  However, as it was mentioned in this chapter, misconceptions, unlike misunderstandings, usually need teacher guidance to completely get rid of the misconceptions.  Misconceptions are usually more deeply rooted and require certain teaching skills in order to remediated.  I can definitely expect certain misconceptions to appear in my units and I must have at least some vague idea as to how to "fix" them before I even encounter them

I think the main misconception I will run into in my algebra unit are those involving preconceptions.  Since my unit deals with solving linear equations and inequalities, the students are expected to use a lot of basic operations, such as addition, subtraction, multiplication, and division, of not only whole numbers, but fraction, decimals, and integers as well.  One of main preconceptions I believe I will run into is students trying to perform operations without converting to similar forms.  Students might just assume that they can add a fraction to a decimal and get a fraction of a decimal instead of converting one of the numbers so that they are in the same form as one another. 

Another preconception I will ultimately run into is that letters now represent numbers.  The first time I was introduced to variables I assumed it was something specific, like a representation of a line length or something like that.  However, variables do not even have to be a specific number, like in inequalities.  Therefore, students might come into the lesson thinking that x is something specific, or even a specific number which is not always the case. 

The last main preconception that will arise in my unit is that of the equality, less than, and greater than signs.  Most students have only encountered one number (or numbers) being equal to, less than, or greater than another number and therefore, they assume this is the only way in which these signs can be used.  But, in this lesson equations can be equal to, less than, or equal to other equations as well as numbers.  This might turn students instantly off to the algebra being taught because they have never seen anything as far fetched as that before in their lives.

All six of the Instructional Principles for Conceptual Remediation will play some role in correcting students misconceptions about the lessons, but the ones that I believe will help the most in my unit are reciprocity, flexibility, appropriate communication, and constructive interaction among learners.  These are four out of the six, but three of these four are kind of interwoven in my opinion.  Reciprocity, appropriate communication, and constructive interaction among learners all deal with students and the teacher talking to and with one another.  I believe this is most important way for students to understand where these "outrageous" new ideas are coming from and for me to see why they do not get it.  I need to be able to effectively communicate with my students and visa versa.  However, some students do not feel comfortable speaking up in front of the class, which is where the communication between the learners comes in.  I can still hear the concerns because they talk about in a group and I observe, but it's without the stress of being put in the spotlight of the entire class.  Flexibility is equal to the importance of communication because if I cannot be flexible and change my lessons to fit the students needs, then their misconceptions will never be addressed to the extent that they need to.  I think with good student-teacher and student-student interactions and discussions and the ability of me to be flexible with the lesson, the misconceptions should be cleared up effectively.

2. Decontextualization

Decontextualization needs to be renamed.  When I was reading this section of chapter 2, I was immediately turned off by the title of this concept since it seemed so complicated and wordy.  It all actuality though, it is basically guiding students' thoughts so they can come to the generalizations of a mathematical concept and work through their misconceptions on their own, or with some teacher guidance.  The one thing that teachers have to keep in mind about this step is that they don't simply ask students questions.  The questions that are posed are meant to get students to think about what they have been working on.  Like the example in the book about the currency conversions, the teacher asked questions so that the students came up with a general idea or process instead of one specific to their examples.  The best part about this idea is that students learn from their mistakes and misconceptions as well as those of their peers.  It is always assumed in the classroom setting that making a mistake or not understanding something fully makes you less intelligent then your peer that may completely get it.  However, this is not the case and the decontextualization stage emphasizes the learning from errors rather than stereotyping students.  I have always learned best when I don't fully "get" something and I have to have someone help me through it.  But, I was never one to ask (I still am not) when I have a misunderstanding in fear as being viewed as less smart.  I have always wanted to teach in a classroom where mistakes were seen as stepping stones for everyone where all students can learn from them.  This decontextualization part of the concept rich instruction uses high order questions from the teacher to get students to see the error of their ways and then see where they are getting that misconception from.  The best way to learn is from one's own mistakes and the classroom should not be any different.  Another great thing about this step 2 is that students are not encouraged to see just one solution to a problem.  They are meant to witness and learn and use a variety of strategies because math can be solved in a variety of ways.  The teacher should not be the "God" of the classroom and sometimes that's what students and teachers both see.  Granted the teacher may know more, but the students know a lot more than they are given credit for.  Their strategies are just as important to the learning process as the teachers right way.

The only issue is I don't know what questions to ask.  The examples in the book seemed way to easy.  Students are usually not that cooperating so how can I spend the adequate amount of time needed for students to learn this way, but not just listen to silence when I ask a question.  This strategy seems so amazing and useful, but also kind of unrealistic.  I wish I could actually see this strategy being applied to an everyday classroom so I was reassured that it actually works and that students don't just give you a glazed over look when they are asked a question.

This component fits in well with the other components, but it is supposed to.  Students practice an idea and then they are questioned during this stage about the more general ideas about the examples they are practicing.  It also allows teachers and students to actually see where misconceptions are being formed so they can be fixed so no one is practicing the wrong thing.  Decontextualization makes a concept generalized so students and teachers can apply to everyday situations that appeal to students.

The first chapter of this book really reminded me of my entire mathematics learning career.  It mentioned how "simultaneous action" is when a teacher introduces a topic, teaches the students how to perform a skill, and then lets the students practice to learn the skill efficiently (pg. 8).  Almost every, single one of my math classes were like this.  Granted, I learned the material, but a lot of my fellow classmates did not.  This system also got very boring, extremely quickly.  When I caught on to a concept rather fast, I had to do pointless problems with the rest of the class while everyone else tried to figure it out.  Lets just say I usually went into a trance for most of the class after speeding through the problems.  I might have learned the material this way, but I definitely did not get the most out of it. 

The other aspect of this book I recognized was my teachers' attempts to use cooperative learning groups.  I have learned that these groups can be very beneficial to students if implemented correctly.  However, my experiences with them have been poor to say the least.  The group projects and discussions I was forced to do simply made me hate group work.  There was no set criteria and if I wanted to get a good grade, I basically had to do all the work.  I know my teachers tried, but I don't think they were taught how to use this tool effectively.  However, after reading this chapter, the fact that neither of these techniques work by themselves was solidified.

To be completely honest, in school I was always one of those people who thought doing excessive amounts of writing in a math class was pointless and unnecessary.  Math was supposed to be about numbers and solving problems not writing.  However, throughout high school I was forced to do math papers about a problem and journal entries about my solving technique.  I watched as peers started understanding math better, maybe not liking it more, but they definitely were catching on to concepts easier and quicker.  I started to realize that this writing process was beneficial to students even myself.  I wrote a paper about how to solve for the height of a tall building and my process of actually solving it.  To this day, I remember the exact steps of how to solve it.  Granted, I am going to be a math teacher so I should know this information, but I distinctly recall everything about this problem and solution.  This article proves a point that I was slowly coming to on my own; writing does belong in a math classroom.  I think the main thing students should write about is how they solve a problem.  They can write a more formal paper or a simple journal entry, but when they write about the process, they are internalizing the information and that is the most important part.  Through this writing process, students start to appreciate math and become less confused.  Maybe they'll even start to like math! 

It is quite obvious from students’ attitudes towards mathematics that something is going amiss in our schools.  The number of teenagers that actually enjoy math is dismal at best, and according to Paul Lockhart in A Mathematician’s Lament, it’s the way that math is taught that is causing this.  Though I do agree with Lockhart’s theory that students are getting “cheated” out of some of the beauty of mathematics, his solution is just unpractical.  Yes, the structure is too rigid and yes, students should be able to explore more with the content.  However, it is ridiculous to think that (even with the new Common Core State Standards) school districts will be able to allow students to dive right in and discover the beautiful mathematics on their own.  There simply isn’t enough time.  I agree that the curriculum shouldn’t be as strict as it is now because obviously what is in place now isn’t working.  Everything needs to be done at a slower pace than Lockhart is suggesting.  Students should definitely not be force fed material.  I believe they should be able to engage themselves with the material and figure things out on their own; just not be forced to come up with everything on their own.  I would be really frustrated if my math teacher expected me to figure out my entire math knowledge by myself.  So some sort of balance needs to be established, my worry is where this balance actually needs to be.  If we expect too much from students, they might feel that the teacher is incompetent and unable to teach the class, but if we keep the system as it is, nothing is ever going to change.  If schools and math classes can ever find a middle ground between Lockhart’s ideal curriculum and the one that is currently in place, students will be much better off.

Mathematics has always been a sore spot with most students in both junior high and high school. This can make becoming a math teacher a discouraging idea because students may just dislike the educator simply because of the subject they teach. As a math teacher, I am going to have to use engaging lessons to attract students’ interest, as well as make the lesson pertain to them. The use of technology is also very important in the effort of drawing students into wonderful world of math land.

The largest hurdle that I am going to have to overcome as a future educator of mathematics is getting students to enjoy the subject instead of hate it, like most do. One of the more beneficial ways of doing this will to connect the math to their everyday lives. Teenagers are more likely to pay attention and like something that directly affects them already. Instead of using math jargon that most people cannot relate to easily, I can use common words that everyone may have heard before. Also, showing my students that they already use skill sets that I am teaching them will lessen the math anxiety that a majority of learners possess. Instead of them focusing on learning something new, they will simply think of it as practicing old skills. Relating the lessons to their lives is only a step though; students will also have to enjoy the lesson itself. Lectures and homework assignments are not going to cut it. I am going to have to employ math techniques geared towards older children as well as be creative and incorporate elementary ideas to make the classes more entertaining and engaging. With these two ideas in place, students should have more fun in math class and therefore learn a lot more efficiently.

One of the main reasons I have observed that students have math anxiety is tied to the testing anxiety that they already have. A bulk of mathematics classes tests their students on almost every concept. Students typically do not test well, nor do tests adequately show the actual knowledge a learner has obtained. The best way to assess students is with the most variety I can think of. Blogs, quizzes, discussions, projects, and in class work are just a few ways to evaluate how well the students are taking in the content. The more assessments I use in all the different learning styles will give me the best indication for how well they are performing in the classroom.

Technology is taking over the world and there is nothing teachers are going to be able to do about it. Students are always up to date on the latest gadget and app that is available and they want to use them. A math classroom is no exception. Though it may be trickier to use technology in my classroom, as I am not a huge “tech savvy,” I am going to have to incorporate it. Since everything is technology based, it is important in the learning. Calculators will always be there as well as computers and it is my job to use them efficiently and to the students’ benefit.

Now, how does one go about balancing all of the necessities of teaching, such as technology, pedagogies, and content? Personally, I do not think there can be a set plan. It is totally dependent on the students I am going to have to teach and the school and state I am teaching in. One school might be more focused on incorporating technology, while others simply care about getting passing test scores. These differences will totally influence how I am going to have to balance everything. In general, I feel that content will obviously be the most important and technology and pedagogies will be determined by the students that I happen to have. School is about the students, so my balancing skills will have to revolve around them as well.