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).

End of quarter tech update

All quarter I’ve been trying to figure out how to use an iPod Touch in the classroom, when there is only 1 iPod and many more students.  While doing so, I’ve found that my main use for this technology has been using it to record interviews, so I can be attentive to the student instead of busy writing notes.  I have also used the voice recording app to record conversations, mainly about behavioral issues, that I could not attend in person.  In this way, the iPod Touch has been very helpful.

Another similar use for the iPod would be to record lesson, so that absent students have access to the lesson in other forms and they listen to it once they return.  Dragon Dictation would be a good app for this, as well as giving students access to lessons when more visual methods of teaching were presented.

In literacy, or for ELL students, a similar tool would be helpful for students that have a hard time reading, so they could listen to a story to pick up the understanding by using visualizing strategies for comprehension.  Though books should not be completely abandoned with this approach, the student could use the book to reference either during or after listening to the story aloud.

There are many apps that are available for students to visualize mathematical concepts, similar to the ones that Robin showed us for the intermediate and upper elementary grades.  These tools give students access to manipulatives and can help students that are “visual” learners.  Additionally, mathematical apps can help students practice mathematical concepts that need the extra practice.

Students today seem to be over obsessed with technology and video games, and introducing technology they are comfortable with accesses their interests and engages them to the point that they may not realize they are learning.  This became evident during fall quarter while interviewing a first grade student about math.  We approached the math problems using video game characters in story problems.  After the interview we thanked him for working with us on math and he replied with, “we weren’t talking about math.  We were talking about video games.”

Though I have used (and found uses) for an iPod Touch in the classroom, I still struggle with the question: When I use the iPod with use a few students, how can I make it seem that I am being fair to the rest, since the others aren’t necessarily using the technology?  The students in my class seem to focus on being treated fairly and it seems like this would be an issue to the students.

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.

Tech blog update

Since my last update, I have not introduced the iPod Touch to the classroom as a way to differentiate instruction.  One of the reasons for this is that I don’t have complete knowledge of the full capabilities of the iPod yet.  Additionally, I believe that some students that are not offered the opportunity to work with the iPod Touch would feel that they were being excluded and I was not being fair to everyone in the classroom.  In 3^rd grade it seems like the teacher being fair, especially with things that are considered a hot commodity (such as working with technology), is very important to the students and they will explicitly express their feelings about fairness (though they may not practice what they preach).

I still believe that using the iPod Touch in the classroom to differentiate instruction is beneficial, but the teacher needs to be comfortable and knowledgeable with the technology to use it to its full potential.  It’s my goal to become more familiar with the features of the iPod, so that I may give students additional tools they need in order to succeed.

Is it possible for a teacher, that may be lacking additional support from either volunteers or EAs, to work on differentiating instruction using technology when the technology is hard to share with the whole class at a time?  Especially when it comes to students needing to feel that the teacher is fair to everyone, how does the teacher incorporate small items like the iPod touch?  I can see that giving all students access to iPods simultaneously would be beneficial, but with the tight budgets in many school districts, this may be a fantasy.  Or, is it meant for teachers to find the potential benefits and be advocates for students’ needs, so they will go out and find the funding through grants (similar to UWB’s approach to obtaining iPods for their students)?  If so, would a teacher want to go about this for their own class, or should it be a school wide effort, so that all students in the school can benefit?

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.

Moving to Assessment

In the past couple of weeks I have been able to see C as both a writer and a reader.  As mentioned in my analysis of C’s writing, she has shown strengths in her interest and ability to write (both from the creative and conventions aspects).  As a reader C has demonstrated that she is at the instructional level for a 4^th grader.  Though she makes some mistakes in her reading, most of those mistakes insertions (adding either “a” or “the”) or substitutions and retains meaning in the text.  C has demonstrated that she is able to summarize the text, and usually points out main characters and major details in the retelling, in her personal reading, classroom assigned reading, and in the readings that I bring to our meetings.  In comprehension, though, C seems to struggle a little with some of the implicit or inferred questions even though she can summarize the passage and answer explicit questions with ease

Taking all of these factors into consideration, I believe that C would benefit from practice and lessons that revolved around reading comprehension.  Since C has expressed interest in reading and writing about animals and fantasy situations, using this as a basis of a lesson on comprehension would be ideal.  Finding a passage about any or all of these subjects will engage C in the lesson and the personal interest may help show her true potential to comprehend reading and ability to answer implicit questions.  As part of the lesson, C could be shown strategies of writing in the text (or making notes) to remember main point or questions that she had while reading.  If the lesson goes better than planned and C demonstrates the ability to answer implicit questions, slowly transitioning to passages that do not involve these subjects may show true improvement in these skills.

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.