Mathematics as Sense Making

Today in EMTH we learned about Sense Making, or how a teacher knows when a student is making sense of what is being taught and how to encourage it.

We started by working on an activity that we had been eyeing at over the past few classes.  It had to do with hand shaking, and what I understood from the question was, that if a party was only attended by couples and some people shook hands, how was it possible for everyone to have shaken a different number of hands?

So, my partner and I began to work on it.  Firstly, because of the vague question, we established a couple of assumptions, such as a person may not shake hands with him/herself, and we also tentatively added that a person likely won’t shake hands with his/her spouse.  As a class we had also decided to begin with a party of 10 people, to more easily find some answer after which we might move backwards towards an unknown number of people.  We created a graph to find that with these stipulations, it was not possible for everyone in the group to have shaken a different number of hands (ex. for a group of 10 people to shake a different number of hands then one must use the numbers 0-9 considering that one cannot also shake their own hand, but it is not both possible for one to shake everyone else’s hand [9] and for one to shake no ones [0]).  After that we removed the assumptions to create a possible answer, but decided that there should be something better (it might be awkward for one to try shaking his/her own hand).  It was then that I realized that we had assumed a stipulation without writing it: each person was only shaking another person’s hand once.  Well, after that my partner and I were able to create possible scenarios in which both the original criteria (don’t shake own or spouse’s hands) were met, and everyone shook a different number of hands in total.

The activity showed more than anything that students come in with presuppositions, and so when a teacher tries to teach something more figurative or abstract, then a student might have difficulty wrapping their head around the foreign concept.  So, for a student to be able to make sense of a topic, one thing a teacher might do is put it in more familiar terms, like using a real life example.  When math remains abstract, not only is it difficult for many people to understand it, but because of their lack of understanding it becomes increasingly irrelevant to their lives, and then what is the point of teaching math if students are not going to use it?  There will always be some math that is not applicable to the everyday lives of a lot of students, but when a teacher explains math in a way that students understand, they will be able to apply it to their lives more, and maybe even increase their overall capacity for understanding.

That is the kind of teacher that I want to be.


Rethinking Assessment

What purpose does assessment have in our classrooms?  And in what form does it take?  Is it for the sole purpose of establishing grades, encouraging students to learn and recite in a particular to get a particular mark on a test or exam?  Does assessment always have to be in the form of an exam?  Or, is there a way to make use of assessment that benefits both the teacher teaching and the students learning?

Recently I read two articles in regards to assessment: Learning to Love Assessment by Carol Ann Tomlinson and Chapter 6 (Assessment) from Our Words, Our Ways.  Both articles talked about how assessment needs to be rethought of to more wholly assess how a student is doing within a specific curriculum, and gave suggestions as to how to change assessment to best benefit the student.

Our Words, Our Ways was focused mostly on teaching students from an Aboriginal background, but the ideas it shares can be applied to all sorts of students.  The chapter talked about how what is being graded should be related to course.  For example, if a student is writing an essay about how the ending of WWI led to WWII, then grammar and punctuation should not take a prominent role in what is being marked, as writing grammatically correct should not be the goal in that situation, knowing the content and making connections between the events therein should.  The chapter was very helpful in emphasizing that just because a student may not do well within a specific assessment style, it does not mean that the student does not understand the subject matter; assessment should have many forms to more wholly assess the student’s understanding.

I really appreciated Tomlinson’s article Learning to Love Assessment because she talked not only about what needs to change within the ideas of assessment, but what thought processes she took to come to that conclusion.  I like how she pointed out that she started thinking assessment was to test learning, which led to thinking that assessment was for the purpose of learning, and finally concluding that assessment is learning.  Her insights into how to use assessment to most effectively teach the students are very helpful.

I want to be an effective teacher, and so that means that I need to use effective assessments to find the best ways to teach and generally encourage learning in all my students.

How does one teach Problem Solving?

Today was a bit different, but very practical and necessary for our learning as teachers.

Today we learned about the Saskatchewan Curriculum Guide, naturally focusing on mathematics, as well as how to use it to then formulate class lesson plans.  We have also been talking about it in our ECS 300 class, which I have found really emphasizes how important this guide is.  The reason it is so important is that it contains the curriculum for every class that can be taught within the province of Saskatchewan, and more importantly the outcomes within each curriculum.  An outcome is a provincially set goal that all teachers across the province are expected to teach their students.  With the outcomes come indicators, which are suggested ways in which a teacher might teach said outcomes.  I have found this learning absolutely indispensable, as this is what we as teachers are being paid to do.  This is the core of our jobs.

With these outcomes and indicators, we have been taught to create lesson plans.  A lesson plan is literally that: the plan of a lesson.  In each lesson plan there is a basic structure.  Firstly, one must define what he/she intends to teach, what outcome that fits in with, and what indicators he/she will use to know it is being taught.  Secondly, one must be able to bridge what has been previously taught to what will be taught, to give context for the lesson plan.  The teacher must then develop a pre-assessment, some sort of device to test where the students are at.  After this, there must be the lesson plan itself, with the hook or set to introduce the topic, the development in which the teacher elaborates on the topic and teaches the content, and then the conclusion in which the lesson is completed.  Interspersed in this must be designated activities for both the teacher and the learner (i.e. complete a worksheet) and approximate time limits to keep the lesson on schedule.  After this, the teacher must have a post-assessment to see if the students learned what was just taught.  A good finish always has a summary of the lesson; it allows the students something short to wrap up their learning and remember the key points of what was being taught.  After the lesson is completed the teacher then needs to see what worked and what did not, so that the next lesson will be better than the last.  Not all lesson plans look exactly like this, but they follow the same basic structure.  Other things a teacher might add to the list are classroom management strategies for if students are becoming distracting or things of the like, or what resources will be used and how each one might help specific learning styles (eg. using a YouTube video, which both breaks up the lecture and stimulates visual learning).

This seems lengthy, but it is organized, and it allows new teachers a place to begin in their teaching, as well as getting them familiar with the curriculum.  Overall, I like it, though my response might be different once I actually attempt to write one.

Have a great day!

Problem Solving: Why is it important to Mathematics?

Today in our EMTH 200 class, our teacher began by asking us why Problem Solving is important, specifically in regards to mathematics education.  We gave various answers, such as bringing up that the textbook stated that when one solves problems within math, then they are more likely to understand math, which increases ability within and excitement for math.  Another person stated that learning how to solve problems in math is a practical skill that can be taken outside the classroom and into the lives of the students, changing their thinking for the better.  The teacher put these two thoughts together and said that the importance of problem solving in math is that it teaches Mathematical Thinking within students, or the ability to think critically about a problem and solve it based upon previous teaching.

We then worked on a couple of examples in which we needed to either use old teachings or come up with new patterns of thought depending on the problem that faced us.  For example, we were asked to create a magic square, a 3×3 square in which each number from 1-9 is used once and only once, and whenever one added the 3 numbers in a row, column, or diagonal they would reach the same value.  To do this, I first began guessing at what the value might be, because then I could determine what groups of numbers would have to be together.  After finding a number that would work, I then created a square in which the columns and rows added to the same value (15 in this case), but the diagonals did not (I originally misunderstood the assignment).  After being corrected, I noticed that another student was creating all the possible groups of 3 numbers from 1-9 that add to 15, and so I did the same thing.  I was able to use the groups from my original square and then add in a few more that I did not originally have.  Then, looking at the amount of groups that a number was a part of, I was able to determine its position in the square.  I had originally thought that the middle number would have to be important, and so I thought it would be an extreme like 1 or 9, but those numbers did not lead to a diagonal addition, but after creating my chart I noticed that 5 was repeated 4 times, and would have to be a part of 4 separate groups, which means that it would have to be the central number.

Here is an example:

Taking this EMTH 200 course has been very enlightening so far, as I get to observe student behavior as a teacher, but I still am learning like a student, and get to make changes as a part of both those sides.  I am excited to keep learning more about students in general, and about myself.

Problem Solving: What is it?

I began my EMTH 200 course this week.  This is a course focused on how to teach math, and how problem solving is essential to one’s own learning of math.  As of yet, this is a very interesting class, and I hope to learn lots that will help me in my endeavors as a math teacher.

Our teacher had us as a class do a couple activities in which we would see a problem and try to solve it based on what we already knew.  The first example was this: if you buy an item and a store and get a 20% discount but have to pay 15% tax, which would you prefer to pay first.  Immediately my mind began moving, but then as I started writing down my theory to contrast whether one way was ultimately cheaper than the other.  The way I would have implemented it would be to multiply the unknown number by 0.8 (20% discount) and then by 1.15 (15% tax), and then the reverse order, after which I would see which number was smaller and select that as my answer.  The only problem was that the answers were the same, and I became confused.  Others in the room received similar answers and had different responses to that.  It turns out that when only multiplying numbers, the order does not matter, and the answers will be the same.  It was afterwards revealed by the teacher that it was an exercise to examine how different students would implement problem solving, and then how they would respond to the results of their inquiries.  I found it very eye-opening.  We did another activity and it was interesting to observe as a pre-service teacher, us the students.

It will be interesting to see where this class takes me, but I think that it will be very enlightening too.

The Struggle of Problem Solving

Today’s EMTH class was interesting, and with the readings we had to do it made for a very thought provoking day.

Today in class we did another activity in which we had to represent the fluctuating body temperatures of a fictional sick woman over a period of time.  Without even thinking about it, both groups decided that graphing was the best way to represent it, and we began working on it.  I found the instructions given a bit vague and confusing, but that only made me want to understand it more and push through the difficulties I was having.  Our group finished it and as a class we discussed it, but I am still think about it.

Later, I was working on my assigned readings and read that it was actually good for students to engage in a bit of a struggle in the pursuit of an answer.  When students struggle through math, the come out understanding the content better, as they really had to engage with it instead of flying through a problem.  When students are easily able to gain answers, they may not actually understand the content, which will be much likely to stick in their minds when they need it, but those that do struggle with and engage the content have is much more solidly planted within their memory, and can more easily recall it when they need it.  I can definitely attest to that fact.  Last semester I took two math courses, those which I will label as my easy and hard math classes.  I had a lot of prior knowledge in the content in my easy math course, and so had a fairly easy time adjusting at the beginning of the semester.  In contrast, my harder math class left me struggling to understand the concepts in class better.  But, as the semesters went on, there was a significant change: my easy math class became more difficult as I struggled to keep up with the new content that my teacher was teaching, while the hard math class was a lot easier to keep up in because I was working out the concepts fairly regularly.  By the end of the semester, I was much more confident in the content that I had engaged in and struggled with than the content that I had not done so with.

So, after learning more about this, as a math teacher I am going to let my students struggle with the content.  Just giving them answers will not help them, but giving them guidance as they find the answers on their own will be much more effective in their overall learning.

Responding to ‘Learning From our Student’s’, by Nel Noddings

Hey everyone, I’m back.

As a quick update, I am now in my second year and fourth term at the University of Regina, and I have been learning a lot.

For my ECS 300 class, which I have started this semester, I had to read Learning From our Students by Nel Noddings.  The author briefly discusses the difference between subject matter and learning, and how students should be pushed to learn what they love, and should not be forced to take subjects that are only an ‘obstacle’ to their goals.  For example, a student who does not enjoy math, who is not good at math, and who does not need math to succeed in whatever field or occupation that they intend to pursue once they have finished high school should not then be forced to take math classes in order to be able to graduate.  The author also talked about how school now leans more than ever towards being in pursuit of the best grades or the highest GPA, and because of this ‘teaching to the test’ has become increasingly prevalent.  Because of this, learning for the sake of itself has been lost, with detrimental consequences for students and ultimately society.  The author concludes that teachers should listen to their students to find out how best to teach them and how best to emphasize their gifts and passions.

My initial thoughts after reading this were very much in favour of the article.  The idea that students should be pushed to increase their learning in what they are passionate about is a very high-minded ideal.  The fact that students are not learning for the sake of learning, but to get good grades and that teachers are expected to teach to the test in pursuit of said grades is quite sad.  I can speak from my own experience that there have been many times that I have gone into classrooms in pursuit of a good grade and nothing else.  Students should be encouraged to pursue what they are passionate about.  But, the more I thought about it, the more I thought about the purpose of mass, public education.  I’ve heard that for some, education is not intended to be for the benefit of students, but instead for the benefit of the state.  Having a educated population is good for economic, political, cultural, and many other reasons, but the purpose is for the benefit of the state.  So, I thought to myself, if the purpose of education is for the benefit of the state, then shouldn’t students be pushed to learn things they do not necessarily want to learn, if the state thinks that it has value?  On the one hand, I don’t think that pushing knowledge down the throats of unwilling victims is a good thing, especially when they will only forget once they have finished the final, but on the other hand we, as students and teachers, are only here because the government has thought it valuable to educate its population, so as subjects of the government, should we not respect that?  It is a difficult question.

What do you guys think, should school be for the student or the state?  And how does learning fit into this idea of school?  I’ll be glad to hear your answers.