Math Blog Final Reflection

All quarter we have learned about different manipulatives to use in our classroom.  Though intended for upper grades (5-8), these ideas for teaching math can be used in any grades (or variations of the ones shown during class).  Using these tools in the classroom will help students understand the “why” and “how” of concepts, which will help them find the importance, remember, and use them.  If not using manipulatives for whole class discussion, they can easily be used to differentiate instruction for students that are need additional practice, or for enhancement for students that need additional challenge.

I have already begun to use random manipulates in the classroom to demonstrate concepts, specifically on the lesson about fractions of different sized wholes.  In this lesson, I wanted students to understand that having the same fraction size, but a different sized whole, will not be equal.  To demonstrate this to my third grade class, I used two methods to demonstrate this concept, first with three different sized bowls and told two people they could each have half a bowl of pretend ice cream.  I then posed the question of if that was fair, in which students replied no.  Then, I said, “but they each get half a bowl” and students showed, using the bowl, how the halves of different sized bowls didn’t mean the same amount.  In the second situation, I used clear Starbucks cups and filled each size half way.  I then had students pour their halves into clear measuring cups to see that the halves were definitely different amounts.  Though this was not a manipulative like the ones that Robin had shown us, it was an object that students could use to see the concept and find understanding with (opposed to the books demonstrations).  I followed up with having the students create their own problems to assess their understanding and application of this concept.

Now having used manipulatives myself and seeing them used in the classroom, I can see that the use of manipulatives is helpful in teaching math concepts. They can teach the same concepts that the curriculum book wants them to learn, but even in less time and with more impact on overall learning.  Knowing this, I will be more aware of how I will teach math in the future. 

Math Blog 7

What did I learn?

During class we had many discussions about various aspects of teaching math.  Of the many things that we discussed was the idea of begin frugal in the classroom and doing investigative work to find classroom resources that will help enhance students’ math experience.  For examples, Robin told us about the projectors that she was able to have when the school switched to doc camera and the computers she was able to obtain from small talk with someone whose company was updating their computers.  While this is good to know for new teachers, it is something that all teachers need to consider, not just math teachers. 

Another thing that we talked about was the mathematical teaching cycle:

All lessons need 4 things (Mathematical teaching cycle): 1. What do they know? (Rationale) 2. What do I want them to know?  What kind of math do I want them to know? (Objective) 3. How do I get them there? (Lesson)  4. Did they get there?  (Assessment)

Though we had covered the major concepts in other programs within the program, it was nice to have such a concise explanation of things that teachers need to have in mind.  At this point in the program, it’s nice to know that as long as we know these major things, we do not have to write out the long lesson plans for every single lesson that we teach, saving a lot of time and energy.

In addition to these things, I learned about tinkerplots and wolfram alpha websites.  Tinkerplots would be especially helpful for my main placement, since students are beginning to work with data.

What do I still have questions about?

How can we, as teachers, have the influence on students that math is important and being good at math isn’t a “stigma”, when parents may be influencing students the other way?  Robin had mentioned that some cultures don’t value girls being successful in math, and believe that males should be the ones that we give focus to, but how do we instill the values of math in every student?  Could we be angering parents with our intentions by doing so?

What are the implications for classroom practice?

Of today’s class, I will most likely use the mathematical teaching cycle as a short lesson when planning all of my lessons, not just math.  This is a quick, easy, to the point tool to use to make sure that everything is planned and accounted for.

Another thing I will use with my class is to use tinkerplots, and other tools, for students that are struggling to understand concepts.  This will be a way to differentiate instruction.

Math Blog 6

What did I learn?

We began this class by completing an activity where we were fitting quadrilaterals into a shape.  This activity made me think about the various ways that different quadrilaterals can look and be altered to look like each other.  It was only with prior knowledge of quadrilaterals that this activity would be easy to complete, but in a classroom this type of discovery could help students remember the various properties of quadrilaterals.

What do I still have questions about?

What other websites have similar manipulates?  Are there physical manipulatives that would produce similar lessons?  I can think of the peg board that a use rubber bands or string, but that would be difficult to produce the same results.

What are the implications for classroom practice?

I have to admit that I had a tendency to playing with the online manipulative, instead of listening to the teacher, so I know that this is something that would have to be addressed in the classroom.  I would need to set clear expectations, clear time frames, or group work to address this.  I’m sure this is something that will be a problem no matter what, since students tend to be so fascinated with the use of technology that they just want to “play” (especially in the younger grades).

Math Blog 5

February 7, 2011

What did I learn?

At the beginning of class, Robin had mentioned some small things that can help us to have successful math classes.  One thing that she mentioned was backwards design, where the teacher designs the test based off of what he/she wants the students to know at the end, then formats the lessons within the unit to match up with the end goals.  Once the draft of the test is generated, then the teacher would edit the test as the lessons progress and the teacher knows how things are going.  I have some questions about this, but that can be found below.

Another thing that Robin mentioned was the use of journals or blogs in the math class. These journals or blogs would be to record the questions that the students are having about the concepts, for students to self-evaluate, and for the teacher to assess the students’ understanding.  This would be done by giving the students open ended questions, so they can fully express their thoughts on math.  While explaining this, Robin also discussed the frequency of having students write in their journals or blogs, which is usually dependent on the grade.

We completed an activity that was designed to have us work with mean and range.  In the activity, we constructed “frogs” from cotton balls and used paperclips to launch them.  We recorded our data after several attempts, then completed the worksheet that involved calculating the mean and range.  This activity did not seem to have objectives explicitly explained before the activity began and the definitions of mean and range were not explained to us.  Of course, we knew what these were, but would a student?  It seems like this activity would occur after explanations or definitions of these words were discussed.

What do I still have questions about?

One of the main questions that I have for this week is about the backwards design of units.  I understand the benefits of creating the final test first, to make sure that you are assessing your objectives/lessons, but the part I am skeptical about is modifying the test as the unit is progressing.  On one hand, the idea totally makes sense, so you are testing appropriately based upon the students’ ability.  On the other hand, as an outsider, it seems like it could tempt the teacher into either making the test too easy and potentially falsifying the students’ knowledge (and a teacher can say “well, my class test scores and grades are high”).  In this case, when it comes time for standardized testing, the students may get lower scores than they were expecting because they had been doing so well in class.  In the other direction, a teacher may end up making the test too hard and it would be more of a challenge than a true assessment of the students’ knowledge.  It seems like the art of making the test appropriate to the class is an art. 

Then, in a middle school setting, would the teacher make separate tests for each period, since each class may be at different ability and skill levels?  Creating a separate test for each class, then grading them seems like it would be a large tasks, but if that’s what it takes to differentiate instruction and to help students succeed, then that’s the role of the teacher.

What are the implications for classroom practice?

The discussions about testing, this week, have given me a lot of ideas about how I would approach testing.  While thinking about it, I tend to think about the application and implementation in a middle school setting, opposed to my main placement in 3^rd grade.  Modifying tests after informally assessing the students’ ability, typing out tests, and drawing pictures to lessen test anxiety are all ways to help students succeed.  I would consider many of these ideas in my own classroom, but another thing I would want to implement is to have students create personal goals a few days before any test.  Setting goals is something that is important for students’ lives, but they don’t seem to have too much exposure to this.  Additionally, having students set goals ahead of time may give them a way to focus their energy and studying and may also give them more ownership and pride in their work.  Hopefully this approach will result in the students learning more, higher test scores, engagement, and potentially a higher interest in math.  Another potential benefit from goal setting will be the students showing more confidence in their math abilities.  To reach these potential outcomes, careful modeling and scaffolding will be need to be done by the teacher ahead of time.

Math Blog 4 - Class Jan 31, 2011

What did I learn?

Today we discussed many different topics about making math assessable to all students.  Included in that discussion was the topic of letting students see the teacher struggle, so they may understand that math is something that can be hard for everyone and that not everyone can understand or answer a problem correctly the first time and that persistence will help them.  For me, I understand the concept and see the importance of it, but the perfectionist side of me will have difficulty.  In preparing for the lesson, I like to have all of the problems worked out, so that I do not waste valuable class time fixing an error (this was especially true in my middle school dyad placement, since time was so limited).  To address problems that caused students to struggle, I would ask students about problems that they struggled with and would work those out.

We also talked about exposing students to language and used the example of moving a 2D object to a 3D object and continually being asked “why?” in order to prove our reasoning.  This class example worked and we came to the correct solution that was supported, but would that be possible with students that do not have prior knowledge of the correct terminology?

Another thing that we talked about was the need for students to have exposure to concepts in order to move them along the Van Hiele levels, but in order for students to fully have the exposure, the teacher must also have that knowledge to help them get there properly.  This idea is one that believe is true, especially because understanding can help teachers become enthused about the subject they are teaching, ultimately leading to the success of the students.

What do I still have questions about?

When completing activities in math classes, such as the mira, GapMinder, and box folding activities, and if an outsider walks into the class (be it the principal, parent volunteer, or other), would they see that as being productive towards the goal in math?  If these people don’t see the value in the activities, especially the principal, could that have negative repercussions for new teachers?  Or, will it be the explanation or resulting test scores that can support the teacher in these practices?

To show students that anyone can struggle with math, as mentioned above, would it be beneficial to intentionally struggle with math and include that in your lesson plan?  Or, should the struggles come organically?

What are the implications for classroom practice?

In the classroom, exposure to math tools is important, but also using everyday items or ideas in order to learn about math is also beneficial.  I found this while trying to show my 3^rd graders the need to divide and having the remainder shown as a decimal (instead of a fraction or R#).  The book only used money as an example, so I wanted to supplement this by using other things.  Besides modeling problems that would need a decimal as the remainder (ex. finding how many miles per gallon, weight), I brought in food containers and had the students work together to find out how many ounces or grams there were per serving.  In order to do so, the students had to find the ounces or grams and the number of total servings.  The students seemed to enjoy this activity and could see real world application to the concept that was being taught.  In the future, I plan on using these ideas, so students can find real world applications while also working hands on.

Math Blog 3

What did I learn?

Gallery walk, the way we see things (and makes sense to us), students may see things differently.  Graph should stand on its own, interpreting other graphs.  Important for everyone to be able to read graphs (create, make inferences, draw conclusions, and justify reasoning).

A new math tool that I learned about was a mira, which is mainly used for teaching about symmetry/congruence.  When I was in school it seemed like these concepts were taught using memorization and direct instruction, so it’s good to learn about tools that will put concepts into action.

What do I still have questions about?

Similarly to my questions from my last math blog, I am wondering if it more beneficial to have students use GapMinder to find information to use (to use real data) or to come up with their own data.  In both class discussion and in one of our Anneburg videos, the idea of having students create their own data will make the students own their work.  Or, like before, is that dependent of grade level?

What are the implications for classroom practice?

In my main placement I would have students create their own graphs, since the information on GapMinder may be too complex for their practical use.  The other thing that we used in class, the mira, would be beneficial for 3^rd grade when learning about kinds of triangles (for example equilateral).  If it was not realistic to introduce the mira to students in this manner, an introduction to the tool would be helpful in an art project, so they can get some exposure to the tool and the concepts of symmetry.

Winter Quarter Math Blog 2

What did I learn?

One of the activities that we did today was to create charts about a change in population, through history, for a given area by using GapMinder.org.  After having an introduction to the activity and website, we were broken into groups and needed to find a world phenomenon and use the graph from GapMinder to help us explain the phenomenon.  Though there were so many options from GapMinder, but it seemed that the ideas that we had didn’t have enough information on the graph to help support the phenomenon that we anticipated, so we had to brainstorm another set of data to use.

What do I still have questions about?

This activity was rather frustrating because it seemed like we could not find the graphs to support the phenomenon that we wanted to present.  If I were to do a similar activity in a classroom I wonder if there would need to be additional boundaries or expectations, so that students don’t get frustrated and become disengaged.  It seems that many students need some requirements to follow, at least with my 3^rd graders.  Or, is this activity (and GapMinder) intended for older students?

When conducting an activity like this, would it be best to have the students (older students – particularly in a middle school setting) become focused by working together with block teachers in order to integrate or enhance other lessons?  Or is it the open-endedness alone that will engage students because they will be using their interests to select data?

What are the implications for the classroom?

Having students find their own data makes the activity more relevant to their lives.  For my 3^rd grade classroom, I would have students find a topic of interest or create a hypothesis before doing research, simply because of the amount of time that it seems to take the students to complete open-ended tasks.  Students at this age can collect data from their peers and graph it, telling a story that is about their classroom.

Winter Quarter Math Blog 1

Today in class, I mostly learned about different ways to come up with the same solution, as we did in finding the functions for the garden.  While I quickly saw patterns and would come up with a function, my partner was able to come up with a totally different way of finding the solution.  In our first problem, which I was able to quickly solve, I did not allow my partner enough time to follow through with her thoughts, therefore, my method deterred her complete thought process.  As we continued working, I knew that I would have to keep my solution to myself for a while until my partner was able to complete her thought and come up with her own solution.  Then, we would show the other person how we came to that conclusion and from there were able to brainstorm alternative methods of finding the solution.

As a teacher, I know that there are many ways to solve the problem, I just need to allow the students enough time to explore their thoughts before everyone share, as to know skew their ideas.  When everyone’s ideas and methods of solving the problem are so different, how do I present the information in a way that will cater to all of those students’ needs (to make sure they understand the information)?  I’m sure that, over the years, I will learn more ways to solve the same problems giving me real examples to use and reference, but what about the first years?  Will the way that I see the problem and present it be enough for every student?  I’m afraid not.

When I do present math in class, especially in my dyad placement, I would often notice that either the language I used or the way I presented the problems was insufficient, making me think about how I could have present the information better, so the students will actually learn the concepts.  When planning new lessons, one thing I strive for is alternative ways to explain concepts or solving problems (which can be found in the modifications section of Jean’s long form lesson plans).  If there are new ways that students come up with solving or explaining, I can then make notes on the lesson plan, so I can use them in the future.

Math IS Everywhere!

Math has always been a big part of my life, but looking back on it, it was probably only because I excelled in it.  In the working world, most of my jobs involved math, especially as an accountant.  It was during my last job that I really noticed just how much basic math and algebra I was using on a daily basis.

One use of algebra sticks out in my mind the most.  At the time I needed to find the base cost of a contract, before fee was applied, and was given the fee percentage and the net contract amount.  After working on it on paper, I came up with the equation:

(gross contract amount) x (fee) = (final contract amount)

1.065X = 1,523,000

*I divided both sides by 1.065

X = 1,430,047 where X is the cost before fee

Fee = 6.5%

Final contract amount = 1,523,000

I use this knowledge in many ways for things in my post-accountant life.  For example, if there is an item that is $15 after tax, I can find the cost before tax.

It was just last week when my mom and I had a conversation about how some people’s lack of math knowledge makes them less efficient in many everyday tasks.  This conversation stemmed from a trip to a store, where the computers were down and they had to figure out prices without the help of their computer.  I had heard the cashier tell the customer in front of me that the tax rate was 9.9%.  In an effort to help out the somewhat stressed looking cashier, I quickly found out the total of my purchase by taking my total purchase and multiplying it by 1.099, giving me the grand total for the sale (I did use the help of a calculator function on my phone). Here is my equation:

5.98 (I did this math in my head because I had two items, one was $1.98 and the other was $4.00)

5.98 x 1.099 = 6.57202

Though I thought I was helping out, the cashier still decided to find the total on his own.  He pulled out a calculator and did the following:

1 + 4.98 = 5.98

5.98 x 9.9 % (used the percentage button) = 0.59202 (wrote down 0.59)

1 + 4.98 + 0.59 = 6.57

Sure, he got to the same answer as I did, but it pained me to see all of the unnecessary steps that he had to take to get to the final answer.  The most surprising point was that he had to add the two items up to get 5.98.  This is a prime example of why I want to become a math teacher.

There are so many other ways in which math is present in everyday life.  I know that I use math when following a recipe, but changing the serving size.  I also use math to figure out how many miles per gallon I am getting on a tank of gas by taking the total number of miles driven on that tank and dividing it by the number of gallons it took to fill up.  Math is also needed to figure out how to give change to a customer, and also in counting it back (which is an increasingly foreign concept).

Throughout my time in the teaching program, knowing that my end goal is to become a middle school math teacher, I have been thinking up ways to engage students and make them realize that math is relevant and will actually be used in their life.  It seems like so many students lose interest in school because they don’t believe they will need to use it after they have completed school. This is simply not true, so I would like to have students tie in math to their world.  One way I have thought about doing this, is to have students find articles on math, think of ways that they use it in everyday, then have class discussion based upon the students’ findings bi-monthly.

My mind was running on overdrive, as I was driving, thinking about all of the school stuff that I had to compete, but specifically about how I use math in everyday situation.  For a split second, my mind followed a song on the radio and I happed to hear the following lyrics from a new song by Rihanna and Drake, called “What’s My Name”:

The square root of 69 is 8 somethin’, right?

‘Cause I’ve been tryin’ to work it out, oh

As soon as I heard these lyrics I started laughing, just realizing just how much that math IS all around us.

Blog, Week 4

This week has been an interesting week, between the classes with student interaction (literacy and math) and in our dyad placement where it was conference week.

Tuesday was full of interesting events, from meeting with our Kindergarten buddy to interviewing first and fifth graders about math.  Our kindergarten buddy literacy session had seemed to go much more smoothly than it had in the past, since we used one of our buddy’s favorite things to approach our literacy activity – Spongebob.  After reading him part of a Spongebob book, we asked him to write in a booklet that we created for him, filling in the blanks “Spongebob is…” and “Patrick is…”  When he thought of a word, we had him stretch the word and write down the sounds that he heard.  He was very open to this approach and worked well stretching out his words, and would write down the sounds that he heard.  We didn’t correct him too much, even when he wanted to write “Spongebob is a cleaning utinsle” as long as he stretched his words and wrote the letters he heard.  Our objective of having him stretch out his words and come up with beginning sounds was successful.

Then, we were on to our math interviews with first and fifth graders.  As soon as my partner and I walked into the first grade classroom we were drawn to a student that was saying, “oh my gosh!” and dropping his jaw as the number of adults in his classroom quickly grew.  Seeing him react this way made my partner and I want to work with him.  When we were told to find a student to work with, we rushed to his side before anyone else could choose him to work with.  Choosing him was an interesting choice, especially once we got into our interview.  At first, when we asked him an addition problem, then asked why he knew that, he said “because my brain knows the answer.”  Then, after asking another problem and getting an incorrect response, he said that his brain wasn’t working right.  After the third problem and receiving a correct response, the student proceeded to tell us that he knew the answer because there was a little guy living in his brain and when his brain wasn’t working correctly, the guy would fix it.  That was not the explanation we were expecting, but it sure was amusing!  The rest of the interview continued to be interesting, as the student talked more and more about video games that he like to play, but most of them were violent and he would say things like, “you want to shoot them in the spine with a sniper rifle, so it’ll be more painful.”  This kind of talk, especially for a first grader, was quite disturbing, but gave us insight to the kinds of things that our potential students will likely be exposed to in their early lives.

Our next interview was with a fifth grader that had mastered the concepts that we were asking him about.  Because these multiplication and division problems were so easy for him, he had a hard time articulating the reasons that he knew them.  I was quickly running out of questions to ask him, and he was becoming rather bored with them, so I had to think of new types of questions to ask him.  I used visual clues around the classroom, to see what they were working on, and then proceeded with new questions about inch to foot conversions and areas and perimeters of shapes.  I was finally able to think of some questions that would make the student think harder and had him explain the thought processes he was going through as he was trying to solve the problems.  Though the student said that he liked to come up with math problems to solve on his own, he seemed to give up quite quickly on the problems that challenged him.

During my dyad placement we had short days because it was conference week.  The conferences were an interesting experience, especially because they were all student lead.  Each student had created a portfolio with examples of their work and also set goals for themselves.  The students had practiced the steps of conducting their conference, but most students did not take it seriously.  This lack of preparation was shown during their conference when they would rush through the material and skip over parts.  When the student led part of the conference was over, we would step in and help them develop S.M.A.R.T. goals for themselves – Specific, Measurable, Attainable, Relevent, Timely.  Many students struggled with this on their only, coming up with goals that did not fit these requirements.  After asking questions about their original goal (using their planner, for example), we mainly found that their overall goal was to get good grades.  We were able to take these new goals and turn them into S.M.A.R.T. goals and use their previous goals as stepping stones in order to reach their goals.

Meeting parents, and seeing their interaction with the students, was an interesting experience.  One example was of a parent that had talked about a student setting their goals now, so they could get into Julliard later on.  The way she said it made me questions whether this was a goal of the student or of the parent.  In further discussions, I learned that it was the student’s goal to go to this prestigious school and her mom was simply encouraging her towards her goal.  Later, there was a mom that kept on correcting her son’s spelling and telling him that he can do better than that.  I was appalled when parents would flat out say that their child was simply being lazy.  If I were that student, I would quickly become discouraged by my parents’ description of my and feel that I should live up to their “lazy” expectations and would then stop trying.  Though this was hard to see, it made me realize how important it is for the teacher to be encouraging in everything that the students do because they may not be getting that positive encouragement elsewhere.