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:

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:

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.

Cognitive Coaching to Student Success

Speaker began by listing his credentials: undergrad @ Aquinas, grad @ GVSU, HS/MS english teacher, then MS assistant principal and then elementary principal.  Presented data from Tri County district showing growth of students in MEAP and pre-ACT (8th grade) growth over 2011-2014.  His argument for such growth comes from administration and teachers being cognitive coaches.

Definition of cognitive coaching- to produce a self-directed person with the cognitive capacity for self-directed acquisition of knowledge.

Key tools that teachers need to use to be the best cognitive teachers include:

Conversation techniques: planning, reflective, identifying problems/solutions, and gathering data on effectiveness of conversations.  Then teachers must have specific mental mindsets everyday to be cognitive coaches; efficacy, consciousness, independence, craftsmanship, and flexibility.  In essence, a cognitive coach believes that all answers lie within the student, and it is the teacher’s job to employ a proper mindset with good questioning skills to really draw out the new thoughts/ correct thoughts from their students.

There exist four supporting functions that coaches do: coaching conversations i.e. “teacher mode”, collaborating- one-on-one students in class, consulting- not something we want to do until the student is basically saying “IDK”, then coaches need to properly/effectively/fairly evaluate both how their students have done with the material, and how you, the teacher/coach has done in presenting the material.

So principals (assistant) serve as teacher cognitive coaches; thus, a.p.’s fulfill the role that checks on the teacher to help the teacher best plan out, execute, and then evaluate an entire unit.  Simultaneously, a teacher serves as their student’s cognitive coach by employing the aforementioned conversational tools to maximize student potential.

Hence a “procedure” for being a cognitive coach both as an administrator and as a teacher is to utilize lots of questions to draw out all of the questionee’s knowledge, and then ask their permission before giving advice/right answer.  This grants respect from the coach to the coachee that is vital to encourage both between administration and teachers, but more importantly, between teachers and students.  So the goal of a coaching session is to get the coach thinking on the same ‘wavelength’ as the teacher/student.

Cognitive coaching as a teacher is a great way to get both formal and informal formative assessments done because each session allows the coach/teacher to get a gauge of how much knowledge the student has.

When teachers are being coached, then the session is about getting their thoughts organized and getting a good feeling  (as an administrator) of how well the lesson seemed to go, and any pros/cons/changes that the teacher noticed pre/post lesson.

He then presented what administrators should focus on during monthly staff meetings- use them as time to better themselves as instructional leaders!!

Discussion of a Lab Classroom cycle in which a host teacher has a coach (administrator from different district) and guest teachers observe a pre-lesson interview between host and coach, observe lesson, then a post-lesson reflective session.  Guests look for all sorts of things including: student comprehension of lesson, issues presented/came up in lesson and how teacher dealt with them, and in pre/post lesson their thinking/emotions/expectations/feelings on the lesson.

Wrapped up with handout Scanned from a Xerox Multifunction Device Device by Jim Knight on cognitive coaching and a spiel on formative assessment:

snowball, thumbs up/down/side, exit slip, etc.

Adaptive schooling is a focus on differentiating between a discussion and a dialogue.  Essentially one directional vs. two directional communication.

Overall, this is what I think of this set up for teachers.  For new/incoming/rookie teachers [defined as less than approximately 5 years experience], this is great because it gives them a new support to bounce ideas off of and a different environment in which they can organize their thoughts and ask questions about their ideas. The only thing I would be concerned about is that my personal preference is that when I ask a question, I expect an answer, not a series of questions designed to make me discover the answer.  Now I realize that I am splitting hairs a bit because that type of questioning is extremely valuable for teachers to use with students, and it can be helpful in this context of having a cognitive coach; however, at a certain point in time, I need to get a straight answer.  Most likely, that would be only a minor thing to be concerned about.

For older/veteran teachers, this set up could go either way depending on how flexible/inflexible they are.  The simplest way to explain this is to consider the old mantra: “no one likes change”.  From there, we can see that those who have been doing things their way for a certain amount of time and then they are told to change it all could result in a disastrous outcome.  That being said, I am confident that those “good” teachers would recognize the benefits of having a cognitive coach.  Then they would augment their approach to teaching accordingly.

As a future teacher, I would really like to have a cognitive coach available to me in my first teaching job because the knowledge of having that extra support goes a long way; plus the coach will be a great source of ideas/advice for how to proceed with a particular unit/lesson.

Presenter: (Steve Johnson)

Blog post 3: Instrumental vs. Relational Understanding

Recently, we, in MTH 329-01, began our unit on proportions and fractions, and a related article is the following one by Skemp.

This is a phenomenal article in which the author makes a strong case as to why all instructors need to begin teaching mathematics, and all subjects, in a relational manner.  His biggest supporting point is the fact that research in related educational subjects reveals that when students are taught relationally, their long term cognitive abilities in that subject are drastically superior to students who were taught the same subject in an instrumental fashion.  Furthermore, Skemp gives four reasons as to why relational teaching (aside from the primary mentioned above) is drastically superior to instrumental:

1. The means become independent of particular ends to be reached thereby.
2. Building up a schema within a given area of  knowledge becomes an intrinsically satisfying  goal in itself.
3. The more complete a pupil’s schema, the greater  his feeling of confidence in his own ability to find new ways of ‘getting there’ without outside help.
4. But a schema is never complete. As our schemas enlarge, so our awareness of possibilities is there-by enlarged. Thus the process often becomes self-continuing, and (by virtue of 3) self-rewarding. (Skemp 8).

Clearly these are all goals that we, as future math educators, want to encourage and see develop in all of our students; however, as Skemp points out with his last section titled “Theoretical Formulation”, these ideals only hold weight in an abstract or experimental situation/classroom, and NOT in actual USA classrooms.  Obviously this is a disappointment, but it is also not the least bit surprising.  This is because Skemp gave four points as to why instructors choose to teach instrumentally.  They are:

1. That relational understanding would take too long to achieve, and to be able to use a particular technique is all that these pupils are likely to need.
2. That relational understanding of a particular topic is too difficult, but the pupils still need it for examination reasons.
3. That a skill is needed for use in another subject (e.g. science) before it can be understood relationally with the schemas presently available to the pupils.
4. That he is a junior teacher in a school where all the other mathematics teaching is instrumental. (Skemp 6).

Skemp also gives some reasons as to why a curriculum or outside forces also push instructors into an instrumental teaching method:

1. The backwash effect of examinations.

2. Over-burdened syllabi.

3. Difficulty of assessment of whether a person understands relationally or instrumentally.

4. The great psychological difficulty for teachers of accommodating (re-structuring) their existing and longstanding schemas, even for the minority who know they need to, want to do so, and have time for study. (Skemp 6).

I would just like to point out that the rise and dominance of ACT, SAT, GRE, Common Core, and the like all dramatically increase the impossibility that any teacher, but especially new teachers, have a chance to even attempt instructing relationally due to the high demands of all the standardized tests and regulated curriculum.  That being said, Professor John Golden told us that the most important thing we can do as teachers is to make the best choices for our students within the realm of things that we, as teachers, can control.  Example being that if our district demands the “butterfly method” be taught to 5th-6th graders, then we as teachers should discover relational links between butterfly method and other fractional topics, thus ensuring or at least increasing [hopefully!!] the understanding of the covered mathematical topics.

Furthermore, I believe that all of Skemp’s reasons as to why teachers forgo relational instruction can be summed up as follows: instrumental is significantly easier and more time efficient to teach to students than instrumental; hence, since USA school curriculums become increasing “big test” oriented, instrumental becomes more and more attractive to all teachers, NOT just new ones.  Additionally, Skemp’s 2nd points in both outside factors and primary factors directly relate to each other, and combined, they create a vicious cycle that creates a never ending need for more and more stuff in a more condensed time frame for standardized tests; thus, leaving only one option available to teachers whom want to cover all relevant material and (due to common core and higher review standards) keep our jobs.  Finally, consider all of Skemp’s arguments for relational.  Upon simple examination, it is apparent that all four look/sound like fantastic goals for teachers to have, but with closer inspection, we notice that a significant amount of effort might/can/will be needed to develop this in every student {approximately 30 per 3-6 classes} equates to 120 or so students that all learn in varied ways.  It follows that, in only an 8ish hour day, that a teacher gets maybe 1 hour per class to “teach”, but that leaves a very small amount of time [even if all 60 min/class is spent in group/one-one discussion] it seems nearly physically impossible to achieve such a lofty goal given current resources/demands on single teachers in USA classrooms.

Personally, I feel that we {as [future] instructors} should endeavor to teach relationally all mathematical content to every single student that enters our classroom for the simple reason that it is our job to not only learn mathematics, but to obtain skills vital to survival and prosperity in a democratic society.  This is perfectly aligned with Skemp’s reasons to teach relationally because, in USA, we tend to value individualism/uniqueness, innovative, confidence, independence, and creativity.  These traits are directly developed and supported from all four of Skemp’s justifications to teach relationally.  This is because a student who internalizes their desire to learn mathematics (and other academic topics) shows a drive to better themselves, and when they discover several different techniques to solving similar problems, this will have built up their creativity and innovative skills.  Also, their skills will grow and the student will no longer need to be scaffolded in mathematics; thus increasing their confidence in both mathematics, and in life in general because mathematics is touted as “hard”, so when the student gains a powerful grasp of mathematics, then they feel emboldened to try other challenging things that may or may not deal with mathematics.

Overall, I really enjoyed reading this article and grasping/processing what the article is saying and how it will impact my teaching methods/choices when I am in the classroom.  That being said, I feel it will be best to try and emulate what Skemp wants, but, until something changes, do not sacrifice material that is on ACT, SAT, etc. in loo of improving relational knowledge on fewer topics i.e. balance between quality and quantity must be maintained, but quantity, in this situation, beats quality.

P.S. We also played a game called “bizz-buzz”, which is an, allegedly, drinking game, which has been adapted to counting fractions and recognizing equivalent fractions.  I found the game fun.  Not much else to say on this.

Thanks for taking a few of your precious minutes to read this.  I hope that you found this useful/interesting.

2nd Blog

Alright, so we are now into the start of February, and I have some new thoughts about proportions.  First I have just finished reading the Billings article on proportional reasoning, and i found it fascinating.  This is due to the fact that she recommends using a “no numbers first” approach to introducing proportions to middle school students.  Her argument for this is that this allows instructors to better understand where their students are in terms of their initial proportional reasoning skills.  Thus this is an excellent way to both administer a diagnostic assessment to gather useful data on our students, but also to use as a formative assessment(s) to gauge how well our students are grasping the concept.

Furthermore, she argues against teaching, initially, the cross multiplication strategy/algorithm used to commonly solve proportional problems.  The basis of this follows from the fact that students will then simply memorize the procedure for dealing with proportions, without necessarily comprehending the logical reasoning behind why proportions work the way they do.

Also, we did an in class activity on Day 7 (1-28-15) in which I had the opportunity to “play teacher” to the class.  For this activity, we were tasked with identifying, in a series of 7 problems, which set of solutions had more “blueness” too them.  What I had to do was illicit student responses and get them to explain their reasoning behind which answer they believed to be right in a ‘pure’ mathematical sense.  Overall, the activity was fun, but as I explained on my thoughts about that day’s activity, I felt like it is difficult to simultaneously divorce your own experience and knowledge from the problem and keep and open mind when dealing with an actual class of middle school students who do not really have a firm grasp on what proportions are.  Hence my solution to clear up some confusion is to be explicit in my questions on formal and informal assessments as to whether or not I am looking for a proportion or an actual integer or something else.  That being said, a good bonus question or formative assessment would be to investigate whether our students can correctly identify which contexts require proportional reasoning compared to integer or other kinds of reasoning abilities.

Thanks for taking a few minutes to read my thoughts on proportions and teaching the topic.  Hope something is useful/thought provoking.

Thoughts Regarding MTH 329 topics

We have already completed 5 days of class, and we have covered a lot of topics in this time frame.  Here are my thoughts regarding some of those topics:

First regarding integers, operations on them, and how to use them for instruction:

Integers form the backbone of all future mathematics that our students will be dealing with as they progress beyond late elementary/early middle school, so how we teach integers is vital to give students any chance at all of succeeding in mathematics.  That being said, it is (to me) best to utilize an instructional method that uses definitions of integers and operations on them in a context that allows for a good discussion of integers and the defined operations on them.  What I mean is that focus instruction on addition, multiplication, and exponentiation [if exponentiation is a middle school topic?] of integers in a relocatable context that survives the odd rules regarding positive and negative integers.  Personally, I enjoy a context that I found while reading on this same topic, and that context is using video games to explain positive and negative integers, as well as addition of these integers.  As for multiplication and exponentiation, I think using money {interest?} or some context that they can easily understand [perhaps rates and labels discussion could be useful like: velocity, etc.].  From there, simply go “by the book” and explain the rules regarding each operation; personally, I think showing students a pattern as to why negative times positive is negative, and then build on that idea to explain why negative times negative is positive [use idea of negatives simply cancelling out].  That being said, to us, negative times negative equals a positive is a fact, but to them, it might be a WTF moment, and this might be an instance that we need to tell them that this is just the way it is.  If there is a “good” explanation as to why this is and it can be translated to a middle schooler’s level, then please let me know and then patent it and become rich in the “mathematical education sense”.

Additionally, we briefly mentioned assessments relating to these topics and the idea of number families.  Number families seem like an excellent way to promote rote memorization of some basic operational results of integers that students can then apply to harder operations regarding more complex integers.  As for assessments, we must all be reminded that our assessments should not just be a “plug and chug” type set up; instead, assessments must go deeper to really measure how well each student grasps both the concept [definitions, etc.] as well as the applications of the relevant math in various situations/contexts.  Personally, I believe using story problems that are funny and realistic [to the students] goes a long way in achieving both of these goals.

Finally, I would just like to briefly mention that everyone of the games/activities we have done in class all have one fatal flaw, and that is time.  Time works against the teacher because we only have approximately 60 minutes or something like that per single class to get a set amount of complex information to the students, and while we might have an entire week/month/tri/se/mester/year to get lots of information to the students, time still ever works against us.  That being said, I really thought the expert/beginner discussion/game/thing we did in Class 4 on 1/21/15 is a brilliant idea that allows the students to converse with each other about ideas going on in the lesson/unit; while, simultaneously, allowing us to informally assess individual student’s opinions/knowledge regarding a specific topic.

Alright, that should conclude this post.  I hope that something is of use to someone if you  took the valuable few minutes of your time to read this.