Effective Math Teaching Practices

Pulling from both NCTM’s principles to actions and teachingworks.org high-leverage teaching practices, I have identified the following areas in which I would like to grow over the course of my Teacher Assisting semester:

  1. Designing single lessons and sequences of lessons- while I know that this will be done eventually in my placement, I know for a fact that I have minimal experience with this skill; hence I would benefit immensely from multiple opportunities to practice it throughout this term.  Specifically, I have a general grasp of how to integrate an entire unit together from beginning to end in terms of the big mathematical ideas {and their applications!} that the students must learn; however the concrete lessons/activities that I would like to use to guide them through the whole unit is an area where I am lacking experience.
  2. Implement tasks that promote reasoning and problem solving- Within the above discussion of designing lessons lies the fact that I do not want to just crank out a bunch of worksheets for students to endlessly work on; rather, I am aiming for rich activities that bridge the gap between the mathematical content and where the students might apply it once they are done with school.  While I have thought of some ideas and observed others, I still want more in order to maximize student engagement and thinking at a higher level.
  3. Pose purposeful questions- while I enjoy asking questions, sometimes it is difficult to know which questions to ask in order to guide the student’s thought process to where I want it to go.  Thankfully, my work this semester in group settings and one-on-one situations enables me to practice and reflect on different methods of doing this to achieve the desired outcome.

Long-term, I want my future classroom to be one in which students are excited to come to because they know/expect/anticipate that the learning they do will be of relevance and benefit to them at some point in their life.

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Change teaching, not teachers

The title says it all.

Reading the book The Teaching Gap and thinking about this prompt, I am struck by the eerie similarity between this idea and the idea of a fundamental shift in how education is done in both America and the world. Let me elaborate briefly.

Since late 19th century, America has adopted the failed public schooling model of Prussia and integrated the evil philosophical ideas of Dewey into both the public (think government!) school system and the universities that train future teachers.  Since then and through today, America has gone through a decline across nearly every conceivable field, so the question remains: what caused that decline?

The ultimate answer is: just as America adopted a bad schooling model from Germany, so did America adopt an evil philosophy from Germany, Kant.

That discussion is far beyond the pale of this post; however please contact me if one desires further readings on this idea. Instead my focus will be to make a more positive case for changing [slowly but surely] the manner in which teachers are trained to teach and how they apply their trade.

Without further ado, I submit that a better way to think about improving both the profession and an individual teacher is to change the mindset from what it is now to thinking about what the purpose and goal of education is; to me, as I am now, I argue that education’s purpose is to allow an individual to learn about whatever topic they so choose under the direction of the educator; similarly, education’s goal is to give the individual the theoretical and practical skills to succeed in whatever path(s) that individual pursues throughout their life.

Given the rise of concrete-boundness, collectivism, and whim-worshipping in the general populace, it is no surprise that my ideals for changing the entire teaching profession, let alone the entire country or even the world are anywhere near success; however these goals can be attained, but a shift in how teachers are trained and how students are taught by those teachers must be done for this to occur.

Again, I intend on writing about this more throughout this term, in conjunction with the class requirements, but please checkout my other blogs & social media for extra information.

Thank you

 

Math Workshops

While reading the first chapter of Minds on Mathematics, I was struck by 2 ideas:

  1. In line with the general shift in teaching away from traditional pedagogy to alternative methods of instruction, the idea of implementing “math workshops” has been proposed as an effective means of presenting mathematical material to students in a manner in which they can better grasp and apply the content.
  2. Math workshops are one of many possibilities available to instructors to create new paradigms in order to provide new avenues for educators to use as they deliver content in the classroom.

What is interesting is one facet that I have not yet noticed discussed in the literature, and that is how to connect mathematics to the current and future lives of the individuals who are learning the subject.

What I mean by this is mathematics has typically been taught as, ultimately, having one of two meanings: a Platonic meaning in which mathematics is derived from another dimension [a mathematical world of Forms] or an interpretation similar to Herbert’s in which mathematics is nothing more than a game in which symbols are manipulated without any real meaning or purpose.  Contrary to both viewpoints, I hold that mathematics deals with this reality [and no other!] and has a definite purpose.  That purpose is to inform individual humans about the world/universe in which they live.

Over the course of these future blog posts, and consistent with the requirements of this class, I intend on developing this idea as it relates to the various topics that I will discuss this semester.

Returning to the concept of math workshops in this context, I argue that they are a good tool to have in the toolbox of a quality instructor; however, math workshops are but one tool to achieve the only valid end of teaching math (or any other subject!).  Hence the most proper way to think about math workshops is but another means to convey this beautiful yet complex information to students.

As I read this book, take more classes, experience {first handedly of course!}, and think on all the various methods available to transmit mathematical knowledge, then I will be better equipped to both teach and convey thoughts on mathematics, teaching, and the like.

Thanks for reading!

A bit about me

My name is Matt Rousell and I am currently enrolled at GVSU in Allendale, MI.  I am entering my final year of course work @ GVSU, and this semester I am Teacher Assisting as part of my training to be a certified teacher in GVSU’s College of Education program.  I aim to become a high school math teacher; however I am sufficiently knowledgeable to teach American and World history, gov’t & politics, Physics and Chemistry and I am interested in all of these topics.  I also enjoy reading books, discussing various topics, and playing games [video, board, or card].

I will be using this next series of blog posts primarily in compliance with the requirements of one of the classes I am taking this semester, but I will still aim to make them as relevant and intellectually stimulating as possible.

Blogpost #6

Let us consider the following question commonly raised by students taking mathematics courses at any level, but primarily I have heard this from high school students:

“How is this going to be used [helpful] in my life”?

Sadly, there is no really good answer to this question.  At least one of the reasons why this is the case is because, for a vast majority of them, they will NOT be using most/any of the math that is being taught in classes i.e. geometry, calculus, complex analysis, complex algebra on trigonometric, logarithmic, etc. functions, etc.  With that fact in mind, what teachers need to emphasize {and this is really hard both to verbalize and for students to really grasp} is the critical thinking skills/problem solving ability combined with the logical formulation of mathematics that all students should get from taking mathematics.

What I mean is that it is true that 90% of students will not be taking partial derivatives of trigonometric and other ridiculous things in mathematics for the purposes of their day-to-day occupation/existence; however, the rules behind how/why mathematicians are able to do those things i.e. the proof and use of logic to define systems/branches of mathematics will be used by them.

First, they must use logic aka common sense aka rational thought in their everyday lives to get and maintain their job and present/conduct themselves in a professional manner that, usually, impresses their superiors.

Secondly, by understanding how mathematical proofs operate i.e start with an assumption and proceed using definitions and other proven results, they can apply that to everyday arguments, discussions, and general decision making for themselves and within their jobs.

This raises a new question for teachers that they can and must answer

“how can I [the teacher] make them [the students] understand this”?

My best answer is to focus less on lecturing and spend more time on both applications of mathematics {patently obvious is physics, but economics works for some as well} and showing examples of logic problems that occur on a daily basis in “the real world”.  From there, teachers can begin to make the case for mathematical proof and how that is helpful to make students better overall writers, and by emphasizing how the logical underpinings of proofs can be extrapolated to legal and nonlegal arguments/contexts.  This should peak lots of student interest because they can start thinking about clever and logical ways to make and win arguments with both their peers, and {unfortunately} their parents.  CLEARLY we, the instructors, are NOT encouraging arguments/conflicts between our students and their parents; however, this will be a direct result of emphasizing logic and its applications in day-to-day life when we use arguments and proof as two huge examples.

Furthermore, instructors can try and shift from a traditional style to a self-discovery style and see how well students enjoy actually learning, by/on themselves, the mathematical content and its potential applications.  There are lots of risks to using self-discovery, and a huge one is the dual required presence of a text that is designed to encourage self-discovery and a teacher who has the skills to translate what they read to “its ‘real” meaning.  Plus, this does not even mention the practical limitations imposed by CCSS, budgets, and manpower conflicts present in most districts.

All that being said, I truly believe that making students recognize the true benefits of mathematics as it applies to their daily lives yields so many benefits; unfortunately, there are just as many hurdles that can prevent teachers from achieving this wondrous ideal.  One final area that teachers can direct student attention to for application is technology/programming that is used in computers/internet/etc.

Overall, I want everyone to recognize that while these options are available and that they appear to go a long ways to closing the gap between “abstract” mathematics and “the real world”, it must be recognized that none of these are a “silver bullet” solution; however, it is obvious that continuing to teach mathematics the same way as it was 100+ years ago is a certain path to failure.  Hence all the time constraints, logistical, and other issues must be recognized as existing, they must be dealt with to allow these needed changes to take place in math ed.

Anyways, that about wraps it up.  In short, students have serious complaints about the usability of mathematics in “real life”, and it is on the teacher to provide an answer to their complaint.  The answer can be found by looking @ the logic underlying all of mathematics and the applications of mathematics in physics and technology and economics {money rules!!!!!}.

5th Blogpost

Here we are nearly at the end of March, and we have been covering a large quantity of topics in MTH 329-01.  There is one  concept that I have come across while in MTH 329-01 that I feel the need to discuss, the idea of group work and collaboration in regards to the “teacher-student relationship and classroom environment”.

The basis for this comes from, primarily, two sources, my own experiences and the following:

https://www.youtube.com/watch?v=b-bJoapxS5o&list=PL1eYMNcRIC5k901YDPauiMOcCl47hIqIl

In short, the “thesis” that the presenter puts forth equates student seating choice in the classroom with how well they perform or how attentive and engaged the student is in regards to the mathematical content being taught by the source of that knowledge i.e. the teacher.  The presenter then goes on to discuss how she combated the mindset of some students that “sitting in the back allows I [the student] to be non-involved with the learning process” by showing how she incorporated various technologies {clickers}, working in groups, and a more active/diverse approach to involving students’ in mathematics education by utilizing more “interesting and applicable” activities.   Personally, I think she has the right approach to getting student’s more active in their mathematics education, but I disagree with the mantra that group work and collaborative effort be the “end-all-cure-all” to help students.  I say this because I dislike group work of all forms; whether forced or unforced, due to the simple fact that in the “big” things in academics {tests} and in life [interviews, actual job, etc.] a person is on their own; hence, it seems counterproductive to encourage a “group mentality” mindset in students currently when they are, in essence, on their own when it comes to some of life’s “biggest” challenges.

Let me be clear though, I am NOT advocating for a child isolationist practice and a classroom environment where no one talks to each other, ever, and the teacher does nothing to engage his or her students, instead, I would prefer to see, at best, a more pro-option approach to student activities in class.  What this means is that teachers must, on day 1, set the tone that their classroom is a safe and nurturing place where it is okay to ask questions, but the responsibility to ask those questions is on the individual students themselves, NOT the group as a whole NOR the instructor.

Furthermore, all in class activities should be done in collaboration with classmates or solo, by choice of each student, but the teacher is equally ready to assist all students with any potential difficulties that arise as they proceed through the activity.

In adopting this approach, I believe teachers can still do their job, students can still show up and learn, but the responsibility for the actual learning rests with the student, not with the teacher.  Clearly, an issue then arises when trying to evaluate how “good” a teacher is, since, in the case of the students who ‘do not care’, will perform poorly on assessments; thus, when a teacher’s performance review is heavily based on assessment scores, that teacher, even if they do a “good” job with every student, and everyone of those students makes their own choice to engage or not, then the instructor is still held liable for the “poor” performance of those students who consciously made the choice to not engage both with their peers and their instructor in regards to the taught material.

The end result of adopting this viewpoint towards teacher-student relationships and the atmosphere in the classroom will be to produce students who are capable, independent individuals who can rationally react to a wide variety of mathematical (and other) contexts and proceed accordingly; while, simultaneously, grooming each adult to be better prepared for “real life” in USA.  I say this because, as a society/country, we strive for completion, independence, and honor the individual accomplishment; hence, the students who become so used to relying on a group to get through mathematical {or other} activities in school will be double burdened when confronted with assessments that they must complete completely by themselves and in a society that demands/expects each person to be rational and to be able to “stand on their own two feet”.

In summation, a powerful balance must be struck between the current trend in all education for “collaborative efforts” and the “traditional” individual work done in academics.  This balance can be found in the form of a compromise where each individual student has the choice to either have more group work available to them or each student can work independently while still having equal access  to teacher support regardless of being in a group or working solo.  The desired outcome of this compromise should be to create students who are more able to express their choices and be aware of the cost/loss of each choice they make while being more ready to deal with a reality that promotes the individual and independence over collaboration and reliance.

4th Blogpost

In the last few weeks, we have focused on the relationship between fractions, decimals, and operations on both of them.  Personally, I feel it is important to start by introducing fractions in a manner that students already have an intuitive understanding of i.e. “one-half”, “one-third”, “one-fourth”, etc. and then introducing the symbolic representation of those words: 1/2, 1/3, 1/4, and so on.  Once they grasp this, move towards introducing concept of equivalent fractions, perhaps using Bizz-buzz or some other format/context, and progress directly to basic operations on fractions.  Finally, we should wrap up this unit by “raising the level” and performing multiple operations on various fractions, both proper and improper, and seeing if students can relationally visualize/explain how/why their answer is correct.

Next, introducing the idea of decimals should follow a similar pattern as fractions, but I would personally skip multiplication and division of decimals because that is almost never done manually nor is it “real world applicable”, plus we just bust out a calculator if presented with decimals anyways, so why not stick with that trend.  Instead, I would like to focus on how decimals and fractions are related [1/2= .5=.50=.500 etc.] and emphasize that pronunciation of decimals “tenths” “hundredths” “thousandths” etc. will really help student’s comprehension of what decimals are and how to write them.  As for addition, subtraction, and ordering of decimals, Professor John Golden introduced a phenomenal game called “Decimal Point pickle” in which students must organize 3 drawn cards (from a standard deck) into decimals and then organize them in proper order.  For more details, see the following link:

http://mathhombre.blogspot.com/2010/05/decimal-point-pickle.html

Finally, we @ GVSU just finished our Spring Break, and I used that time to conduct my requisite 12 hours of observation @ the middle school level and to conduct some student task interviews.  This experience was a good one, and I got to experience more “behind the scenes” activities of what goes into being a K-12 teacher.  A specific thing that was brought to my attention via my discussions with my supporting teacher was that all the changes in educational standards and standardized tests really make their job simultaneously easier and harder.  Teaching becomes easier when you are given a set of objectives and the final test all pre-made to measure final knowledge in a unit/subject, but it becomes harder to both prepare for such a standardization, and then to develop lesson plans, activities, discussion points, etc. that align with the new curriculum such that they also assist with student learning and understanding of the material.  A common complaint I repeatedly heard was the lack of funds/resources and time for teachers to have to both meet these demands, and to provide ideal instruction to all classes of students.  The issue is, no one has any one clear cut solution that adequately solves all of these problems, so they will fester until significant changes (governmental, curriculum-al, sociological, etc.) regarding views/expectations of teachers and education.

Overall, we have brought up several important ideas regarding decimals and fractions and their presentation to students; however, a small sample of field work that I did reveals a growing divide between the ideal theoretical that is discussed in university education courses and reality of K-12 schools.  That being said, I did think of a solution to the time issue in regards to 2-hour delays, snow days, and length of school days/classes.

Currently, schools run from September-June, with a 3-month Summer vacation , so what I propose is to have a 3-month Winter vacation running from approximately Christmas time in December till end of March/beginning of April.  This completely removes approximately all issues of Snow Days and delays, thus reducing burdens on teachers to play “catch up” with material.  As for school length/class length, I think having somewhere around exactly 60 minutes per class with around 5-8 classes (depending on district, state, etc.) with lunch/recess of 30-60 minutes will help the overall flow of the day better for all involved.  Obviously a ton more detail and planning for logistical support would need to be done to implement this, but I believe that this set up could significantly reduce the time crunches that teachers always seem to feel.

Hope this is a useful/helpful/thought provoking post for all of who who have taken the time to read this.