1.5+Daily+Summaries

On this page, an assigned student will write a daily summary of the big ideas developed in class along with any other information s/he wants to share, such as the At Home Extension. Here is a sample format I'd like you to refer to when writing the summary:


 * = **The following ideas were the focus of today's class:** =
 * idea 1
 * idea 2
 * idea ......
 * The way we developed idea 1 (2, 3, etc) was ...
 * An important thing to remember about idea 1(2, 3, etc) is ...
 * Idea 3 (idea developed late in the class session) is something that we'll revisit in later class periods but we got a start on this idea by discussing ...

=
Please try and follow the guidelines above when writing your summaries. Avoid writing a linear "Here's what we did first, then second, and so on". The intent is that you begin to think more broadly about the ideas and find more connections; beginning to think like a teacher! So try it out. ======

SESSION 25: THE LAST ONE! Thursday April 19th Big ideas: • What technology is out there and maintains mathematical, pedagogical, and cognitive fidelity? • What is teacher evaluation all about in Michigan?

The way we focused on idea 1 was to review what's available on the web. We were finishing this investigation up from Session 24. We examined virtual manipulatives on the National Library of Virtual Manipulatives page, looked at activities available through NCTM's Illuminations page, and referred to Jennifer Suh's page that shared online applets across various mathematical topics. In addition, we looked at a few Web 2.0 tools and played briefly with the Livescribe pen, considering possibilities of how such tools can promote mathematical learning.

Our guest speaker was Ms. Lindsay Noakes who currently works in the KPS Administrative offices focusing on Assessment and Evaluation. She shared the former process in which teachers were evaluated (before the Race to the Top, RTTP, moneys were available). When RTTP became available, Michigan wanted to compete for those funds but had no state teacher evaluation plan in place so new laws were passed that changed the ways and frequencies for when teachers and administrators are now evaluated. Ms. Noakes shared with us the current KPS rubric for K-12 teacher evaluation, all 21 pages of it! Quite a document. All of this information is posted in our dropbox for further detail and reference. We examined the current rubric and looked for criteria that surprised us, that we thought were reasonable, and those we thought were unreasonable. Interesting conversations pursued as we found expectations for some "highly effective" criteria appeared to be out of our hands, for example, those that related to students attitudes and beliefs. We can not "make" a student appreciate mathematics; we can only provide experiences and an environment in which that can happen.

We appreciate both Diane and Lindsay visiting our class and filling us in on current information regarding mathematics teaching within the state and the local KPS system.

Don't forget we need to read the article "Understanding Teaching" in the dropbox and watch the video on surface area from Cathy's classroom.

I had a great semester with all of you! See you Tuesday for the final! Bring your computer if you want to complete your final using Word; otherwise, bring a pencil or pen. I'll bring some donut holes. Dr B

Session 24: Tuesday April 17th Wrapped up discussing the different curriculums in the textbooks we looked at. - MiC: Similar to CMP and includes several books with a plethora of questions and opportunities for assessment - CMP: Very organized with practical enrichments (examples and problems) - Math Scape: A lot of problem-solving and plenty of learning modules - Math Thematics: Only textbook with one, all inclusive book, similar to the rest (content) - Overall: All seemed fairly similar, good effort to all cover the same overall curriculum

We also looked at various tools that we can use technology in the classroom, I noticed that some of them were actually pretty fun to “play” or use… Who knew math could be fun?!?

Side conversations - How much can we trust students with technology? Would we be able to trust them with texting in class or using laptops during class? - How can we use technology to create a class community outside the classroom? (Wikispaces, Facebook, …)

Session 23: Thursday April 12th Today we had a guest speaker. We talked about the triangle of student involvement. there were three outside triangles, "Learning & Assessment Expectations", "Feedback", and "Instructional Changes". Between LAE and F we ask "Where is the learner at?" Between F and IC we ask "What needs to be done?" Between IC and LAE we ask "Where is the learning going?" After that, we start back at the beginning.

We also talked about how to pull objectives from the standards and how there can be many objectives in just one standard. We were shown the "I can..." sheet again. We unpacked our own standard (Interpret the parameters in a linear or exponential function in terms of a context).

(I think I am forgetting something, but I don't remember what that something is...)


 * Session 21: Thursday April 5th**

Curriculum, curriculum, curriculum…..

The idea of the day was comparing curriculums; we did this in two different phases.

**Phase 1:** Continuation of analyzing and comparing two particular middle school curriculums, namely **//Passport//** and **CMP** (Connected Mathematics Project). The analysis covered seven issues broken up into seven sections on the given worksheet. These sections were: kinds of questions asked, way in which formulas and definitions are learned, level of thinking and reasoning required, nature of the instruction and class room discourse, view of mathematics implied, context of the problems, connections made between lessons/problems. The class seemed to favor **CMP** for its in depth and thought provoking teaching style, where **//Passport//** leaned more towards the old standby of give examples and ask short/one solutions answers.

**Phase 2:** Class was broke into three separate groups and each sent to a different middle school curriculum to examine for later comparison; the four different curriculums were **//MathScape//**, **//CMP//**, **//MathTHEMatics//**, and **//MiC//**. Each group was given the task of analyzing eight different areas of interest; format/contents of a typical unit/chapter, format/contents of a typical ‘lesson’, nature of instruction that the text lends itself to, kinds of questions asked in the text, opportunities for assessment, use of tools for learning mathematics, resources/information for teachers, and view of mathematics that the text presents. As a class we only got to fully analyze one set of curriculum and briefly start to a second set when class ended.

Session 20: Tuesday April 3rd

I. The Case of Robert Carter Some instructional practices that we discussed in class were instructional decision para. 11, Defining vocabulary para. 9, Students explain answers para. 12-13, Group solution presented by one person para. 20, compare and contrast similar graphs para. 22, Partner pair (groups of 4) para. 22, Observing groups and checking group make-up para. 23, and Formative assessment the next day para. 25. These were some practices discussed, but I'm sure there are more, so if you find some, you can add them to my list, or post discussion topics about them on the home page.

II. Spiral Curriculum Building on the curriculum by teaching similar things, but increasing the difficulty and information gained when discussing it again, but we have gained mostly concentric circle curriculum's, where we teach the same thing again and again without making it any more of a challenge. This is something to keep in mind when we become teachers, is to note the standards and expectations, so we know what was taught and how to build off of them.

III. Curriculum cont. We continued looking at curriculum used by schools, like passport and CMP, and analyzing details about it, like what type of questions were used, what type of learning was used, and looked in-depth at the text.

Session 19: Thursday, March 29th-

I. Debriefing our slope/rate of change lesson at Maple Street 7th graders Deciding whether our objectives were met that we established before teaching our lessons lead to much insight on struggles we will have as future teachers. We both explored how each particular grouping of students had a different dynamic that added to how successful meeting objectives was, as well as what instructional decisions we could have planned differently to reach these goals. It appeared to be a combination of the the two that determined our lessons' turn outs. One of our biggest struggles with the student was how we can manage their behaviors and we decided we must think of ourselves not as "pre-service" teachers but as actual teachers and present ourselves accordingly. One of our biggest struggles with instructional decisions was how to help students best interpret the graphs and draw connections-- we decided that fostering discussions among group members is beneficial to helping students in doing so. We will use these insights as we plan our 6th grade lessons.

II. And what happens when it's a V/S graph? Once again, we were humbled by a lesson targeted toward middle school students-- walking a velocity/time graph. We experienced a moment of "how do we do this.." when shifting away from distance/time that both demonstrated how much our students, as well as ourselves, can learn from using technology in a TPACK-supported lesson: one which does much more than merely calculating answers for us, but gives instant feedback and intrigues students. When we ourselves are forced to think in this way, we see how we enjoyed the challenge it brought us and what conceptual connections we were making; it would thus be hard to argue against this kind of learning for our students.

III. Curriculum At the end of the class, we began looking at curriculum that schools used and began asking the questions of whether particular textbooks had questions that would allow students the most learning. Although we just began our investigation, we can relate these texts to what we have discussed throughout the semester about what "the most learning" is and what it actually looks like.

Session 18, Thursday, March 22, 2012


 * The following ideas were the focus of today’s class: **

1. Discussed lesson objectives for our Slope/Rate of Change lesson at Maple Street Middle School on March 27.

2. Watched portions of a video from a prior Math 3500 class of Dr. B’s.

3. Discussed key points for the reading, “Every Picture Tells a Story”.

<span style="font-family: 'Arial','sans-serif';">4. Reviewed the 7th Grade GLCE’s addressing slope.

<span style="font-family: 'Arial','sans-serif';">5. Critiqued each other’s Slope/Rate of Change lesson plans.

<span style="font-family: 'Arial','sans-serif';">(1) The main lesson objectives are for students to have an understanding of the slope for a linear equation (constant rate of change) and the meaning of a positive, negative, and zero slope in context.

<span style="font-family: 'arial','sans-serif';">Student activities recommended for the lesson were pre-assessments, interpreting a graph in relationship to slope, relating motion to slope, matching a graph utilizing the CBR, writing a story which includes the slope concept, calculating the slope from two points (higher level), and providing a written summary of what has been learned in the lesson.

<span style="font-family: 'Arial','sans-serif';">(2) The clips of the video highlighted some instructor errors we want to avoid in the classroom – funneling answers to students, not waiting long enough for student responses, and monopolizing the conversation. It did highlight the fact that we should encourage students to try to create a vertical motion line with the CBR and discuss the outcome of that activity.

<span style="font-family: 'Arial','sans-serif';">(3) The main idea extracted from this article is that we should focus our lesson initially on getting students to become familiar with graphical representations of distance over time and provide the kinesthetic activity (CBR) to do that. Numbers should be introduced at a later time.

<span style="font-family: 'Arial','sans-serif';">(4) We reviewed GLCE A.PA.07.06 specifically and wrote a listing of “I can” statements which the students and teachers will use to evaluate mastery of the standard.

<span style="font-family: 'Arial','sans-serif';">(5) We paired up to look at each other’s lesson plans and to offer input for improvement.

Session 17 Tuesday March 20
Main Ideas:
 * 1) Today's class was focused largely on our teaching experience we will be going on next Thursday, March 29th
 * 2) Go over requirements for FERP that's due Thursday

(1) Since our next field experience is sneaking up on us next week, class was focused around that mostly. Dr. B first gave us a problem with the "add-add" method, like with the hats and scarves in order to demonstrate to us that even though we did something very recently, sometimes it's difficult because concepts don't always "stick". That might possibly be what it's like when we go back into the classroom next week.

Then, we discussed our experiences we had with our last lesson and went over what to do in times when the students are completely stuck and just aren't grasping what you want them to. You can either word the question a different way (maybe put it in different context), or you might have to backtrack to more simple concepts to see if they understand the fundamentals of the lesson. The main thing we are trying to get them to understand is slope/rate of change, but remember to be prepared in any case, if the students are more or less advanced than what you anticipated. They might get it and move on very quickly, so have some things for them after they understand the time/distance graphs. (Computing slope and/or y-intercept, writing equations, making their own graphs with story problems, etc.)

Hopefully everyone is getting comfortable with using the calculator and Calculator-Based Laboratory (CBL), because we will be using them next week in the classroom. You can either have your students experiment with a blank graph and let them create their own, or have them try to match a pre-programmed one.

You should have your lesson plan complete (with minor adjustments needed) by Thursday. Bring in a hard copy

(2) FERP's are due Thursday. **REMEMBER** to go over the rubric when writing it so you know what you will be graded on, and understand the topics you should be discussing. Use your transcript to refer to in the paper. If you are sending it to Dr. B any later than 8:00 AM on Thursday, email her to let her know and tell her what time throughout the day you will have it done.

__Reminders:__ FERP and lesson plans due Thursday. Also, don't forget to read "Every Picture Tells a Story" but there is no Wiki due. If you have not posted a positive thing you learned about your own teaching in On a Positive Note, please do so.

=
<span style="color: #7c00ff; font-family: Georgia,serif;">We worked more with the Calculator Based Ranger device near the end of class. We talked about letting students work through their own mistakes. We discussed the pros and cons of using technology in the classroom. =====


 * Session 15 Tuesday March 13**


 * The following were main focuses of today's class: **
 * 1. Review/Share Thoughts/Analyze our teaching experience. **
 * 2. Analyzing Teaching Tools (Affordances vs Constraints). **
 * 3. Played with new teaching toys. **


 * We started off with a review/analysis of our teaching experience from our last class session. Focused on how we measured up against common teaching issues, i.e. overemphasis on answers, giving away to much info, not providing time for thought, lack of detail on variation, and lack of detail/care with teacher responses. **


 * Another main topic from class was the analysis of teaching tools, we went offered thoughts about the affordances and constraints of the blocks we used and word problems. **


 * <span style="color: #074d07; font-family: 'Comic Sans MS',cursive;">Blocks **
 * <span style="color: #074d07; font-family: 'Comic Sans MS',cursive;">Affordances: visual, tactile **
 * <span style="color: #074d07; font-family: 'Comic Sans MS',cursive;">Constraints: non-precise, had issues w/ the tools, distracting (play), separate pieces/shapes (creating alternative wholes) **


 * <span style="color: #074d07; font-family: 'Comic Sans MS',cursive;">Context (word problems) **
 * <span style="color: #074d07; font-family: 'Comic Sans MS',cursive;">Affordances: interesting, engaging **
 * <span style="color: #074d07; font-family: 'Comic Sans MS',cursive;">Constraints: irrelevant attributes **


 * The last thing we did as a class was get introduced to the Calculator Based Ranger device in which you use a motion detector and graphing calculator to try and recreate a graph by way of movement. **

Session 14 Tuesday February 28


 * Show up at Maple Street Magnet at 7:25 a.m.**

Continue to work on lesson plan. E-mail Dr. B your lesson plan no later than **tomorrow February 29th.** This is to “force” us to get something down and be prepared before walking into a classroom of middle schoolers. Really start to think about what problems you are going to give to your group of students and make sure you thoroughly understand what each problem is asking to avoid confusion with the student.

Read “Thoughts on Fractions” and have a discussion on wiki page of the //Future of Fractions//. **Due Monday at midnight (remember that’s Monday morning) March 12**

Discussed our thoughts on //Future of Fractions//. Should we teach decimals or fractions first? Should we learn tenths and hundredths before we introduce students to the 1/10 or 1/100 fractions?

Second Reflective Writing; **Rewrite due March 13th** - Go over changes to our papers that Dr. B has brought to our attention. - We looked at the rubric for the paper; go ACROSS each row and not DOWN each column - Talk about Fran and Kevin **//__separately__//**, contrast them, assess their teaching (can blend the classroom-based indicators and teachers actions for each teacher, but keep Fran and Kevin separate) - __Do not need to comment on the “new expectations”__

We watched Cathy’s video with the proof and some other videos to observe how the classroom is setup. How does Cathy set the tone in the class? How does she get the students engaged and thinking about the proof? Does it seem that Cathy is strictly following a lesson plan or is she just running with what comes to her?

Session 13 Thursday Feb 23rd

We looked at Cathy's classroom. Some important things to notice were: wait time, asking students what they thought of each problem, along with having them do the problems to learn the methods (focusing) Asking students what?? , using students to continue what another said so having students listen to each other?

We talked about the 2(n-1) problem, most used area models to show that 2(n-1)= 2n-2. This worked because it is "easier" to generalize using an area model than it is a set model. This problem showed the importance of justification, because we needed to justify our answer and diagram, and this reasoning had to be outside of the procedure. for what? Be a bit more complete in the ideas.

Lesson plans- two groups, each group uses different problems? Which problems, divide across, pattern blocks?

Big Ideas- Lesson plan, importance of planning ahead. Understanding concepts compared to understanding just the procedure. Justifying our answers is important, and is needed for all students.

//**<span style="color: #0a4c8f; font-family: Georgia,serif;">Session 12: Tuesday, Feb 21st **//

<span style="color: #0a4c8f; font-family: Georgia,serif;">By examining the problems that we came up with representing 2 1/2 divided by 3/4, we saw how it isn't as easy as it sounds to come up with a true division problem-- when we thought it was division, it ended up being other things: subtraction, <span style="color: #ff00ff; font-family: Georgia,serif;">(although division is REPEATED subtraction so this may be salvaged) <span style="color: #0a4c8f; font-family: Georgia,serif;">multiplication... When coming up with problems for our students, we have to be careful of how we word things to decide if we are really asking the right questions. Also, in some of the problems, we saw that the way we worded problems did not make it clear that students needed a fraction in their answer because wording lead them to think of whole numbers, so again we need to pay special attention to how we word our problems.
 * <span style="color: #0a4c8f; font-family: Georgia,serif;">Examine closely-- Is it really a division problem? **

<span style="color: #0a4c8f; font-family: Georgia,serif;">**Concepts of Division: Measurement and Partitive** <span style="color: #0a4c8f; font-family: Georgia,serif;">As we attempted each problem individually, we used different methods in solving them. In particular, some would use a partitive, or the fair sharing approach, while others would use measurement (which differed too between an area model and a number line). We established that we must be prepared to understand and explain <span style="color: #ff00ff; font-family: Georgia,serif;">(I would caution with Explain; I'd prefer you need ways to question students so they can explain ideas for themselves. So be prepared to understand and craft questions to help your students solve problems) <span style="color: #0a4c8f; font-family: Georgia,serif;">all ways to solve problems because our students will attack problems in both ways; even though personally I may prefer measurement, I'll need to be able to see the partitive approach in preparation for students who use it.

<span style="color: #0a4c8f; font-family: Georgia,serif;">**Shining Light into why we "flip and multiply": why proof is important** <span style="color: #0a4c8f; font-family: Georgia,serif;">Although the algorithm for solving division problems is effective, we established that it does not provide our students with conceptual understanding. Though our students may not need to be able to write a proper proof for this, I feel safe in saying we agree that they should see what dividing fractions actually looks like by using problems we have been working with and manipulatives. It may also be appropriate to show students the formal proof <span style="color: #ff00ff; font-family: Georgia,serif;">(again, maybe help them to work through it themselves) <span style="color: #0a4c8f; font-family: Georgia,serif;">to take the idea of magic out of dividing fractions-- showing them that there really is reason to why we invert and multiply and how this algorithm came about can eliminate their blind actions of following the procedure without reason.

<span style="color: #ff00ff; font-family: 'Arial','sans-serif';">Session 11, Thursday, February 16, 2012


 * <span style="font-family: 'Arial','sans-serif';">The following ideas were the focus of today’s class: **

<span style="font-family: 'Arial','sans-serif';">1. Identifying whether mathematical tasks are high level or low level.

<span style="font-family: 'Arial','sans-serif';">2. Keeping mathematical tasks at a high level.

<span style="font-family: 'Arial','sans-serif';">3. Presenting, interpreting, and orchestrating discussion when giving high level tasks to students (“Happy Face Cookies” problem, “Batches of Cookies” problem).

<span style="font-family: 'Arial','sans-serif';">(1) We discussed some of our ratings on the “Characterizing Mathematical Tasks” worksheet, which asked us to determine whether particular tasks appeared high level or low level. This task challenged our thinking of what constitutes a high level task. Our determinations needed to be justified. We compared our interpretations to how the authors characterized the tasks based on features present. We observed that all the high level tasks involved multiple steps/actions/judgments and required an explanation.

<span style="font-family: 'Arial','sans-serif';">(2) The focus in the classroom is to assign high level tasks to students and to incorporate an implementation strategy that keeps those tasks high level. If tasks are becoming low level, teachers need to analyze the factors that are contributing to that digression. We discussed the handout highlighting factors associated with patterns of maintenance and decline of tasks. It is the teacher’s job to maintain that high level of learning in the classroom and to be able to recognize factors that may jeopardize that objective.

<span style="font-family: 'Arial','sans-serif';">(3) The cookie problems brought to light how important it is to present our solutions with clear diagrams, thorough explanations, and appropriate mathematical vocabulary. The exercises allowed us to practice explaining our problem-solving techniques and to listen as others explained their techniques.

<span style="font-family: 'Arial','sans-serif';">In order to interpret student responses, we realized that teachers must be active listeners, since some problem-solving techniques that will be utilized by our students may be foreign to us. We have to be prepared to anticipate different mathematical responses and be open-minded.

<span style="font-family: 'Arial','sans-serif';">In order to create an environment of discussion, the teacher must be able to compare/contrast responses, create a safe environment which enables students to want to share their responses, show how student responses can build on each other, ask questions that will make students clarify their thinking, bring out common misconceptions, and allow for students to rethink their strategies and make appropriate corrections.

Session 10 Feb. 14th, 2012: Yesterday we began the class discussion with looking at Smith and Schwan's redefinition of success. What makes a lesson successful? How do you evaluate success? We also mapped out what we've done so far in class this semester, starting with working on it by ourselves then moving on to discuss it as a class. Don't forget to spend at least about 20 minutes on that before class Thursday morning! So far we put "Teaching Middle School Mathematics" as our main, central idea. Off of that, the first branch was NCTM and PSSM, which had the Principles and Process Standards branched out off that. The next branch off of the central idea was "classroom practices, norms, and culture" which had a couple more ideas off of it. After the mapping, we had a long discussion about national and state standards, and the process of turning the Common Core Standards nation-wide. So far, there are no standards that all 50 states have adopted. There are a set of Common Core Standards, though, that 44 (maybe 45, maybe 43, we weren't positive) of the states have enforced, including Michigan, so we're in the process of the whole country adopting them but not quite there yet. The next thing we did was discussed high level vs. low level tasks. We viewed everyone's votes on the list of problems A-P and discussed why they were either high or low level. We decided that high level tasks: If it is a low level task, it The last thing to mention is that we were given a problem, which is two and a half divided by three fourths. We are required to think up a story problem by Thursday morning for this.
 * take time to complete,
 * require an understanding of relationships,
 * have choices to make,
 * have multiple solutions,
 * require interpretation and an explanation of solutions.
 * is basic computation,
 * not much thought on how or why, and
 * is just following a procedure.

Reminder! Reflective Writing is due February 21st!

Session 9 Feb. 7, 2012:

Today we looked at how to take one-third of one-fourth and one-fourth of one-third with the blocks (area model); thinking of why we broke up the “whole” of two hexagons into the partitions with the rhombus, trapezoid, and triangle. Also, we talked about how multiplying one-third and one-fourth is the same as doing one-fourth and one-third; commutative rules and reason for multiplying fractions. - Why do we seem like creatures of habit and subconsciously just know what to do without, at times, a reason for why we do what we do?

We took 12 triangles as a whole and dividing into groups of three and four (set model with the measurement conception of division).

Discussed Cathy’s classroom video “Division of fractions”. A couple call-outs that we picked out were: - Visualizing the problem, making sense of the problem, and proving you are convinced - Trying to come across a process to go through the problem in a way that makes sense to you and to others

writing” second.
 * Remember:** Reflective Writing due February 21st. The reading is in Dropbox in the “Second Reflective Writing” folder. Read “Beginning Fraction Task” first and then “Case study for Ref

Session 8 Feb. 2, 2012:

We focused on language choices in the class room and how word choice impacts understanding. We also discussed that we should anticipate student thinking. The misconceptions and errors that students have on specific topics, is something that we should think about as teachers too.

We discussed how to guide the lessons and the beliefs in the classroom. We went over the take one-third problem and also the "Learning to Notice" article. We all added to a Google docs article Learning to Notice.

<span style="color: #074d07; font-family: 'Courier New',Courier,monospace;">The following were main focuses of today's class:
====<span style="color: #074d07; font-family: 'Courier New',Courier,monospace;">We started off with a quick review of the six NCTM Principles, which are Equity, Curriculum, Assessment, Technology, Learning, and Teaching. This was more a brief review to see what we remembered from our last session. ====
 * 1) ====<span style="color: #074d07; font-family: 'Courier New',Courier,monospace;">1. NCTM Principles ====
 * 2) ====<span style="color: #074d07; font-family: 'Courier New',Courier,monospace;">2. NCTM Process Standards ====
 * 3) ====<span style="color: #074d07; font-family: 'Courier New',Courier,monospace;">3. Fraction Representations ====

====<span style="color: #074d07; font-family: 'Courier New',Courier,monospace;">For the NCTM Process Standards we presented our four Google Docs that we had written in groups of two (except Josh) since the previous session. Altogether there are five Process Standards; Problem Solving, Reasoning, Communication, Connections, and Representation. Our presentations were over the last four standards. During the following discussion we noted that all four standards were interwoven into one another a fair amount. ====

<span style="color: #074d07; font-family: 'Courier New',Courier,monospace;">Process Standards

 * PS || R || C || C || R ||

<span style="color: #074d07; font-family: 'Courier New',Courier,monospace;">Principles
====<span style="color: #074d07; font-family: 'Courier New',Courier,monospace;">We also finished up our review of the 9 representations of ¾’s worksheet. There were four different types of representation; Part to Whole, Division, Ratio, and Contextual with the Part to Whole representations being split up into three model types being Area Model, Set Model, and Number Line Model. ====
 * < C ||< A ||< T ||< T ||< L ||< E ||

Session 6 Jan. 26, 2012:

Remember to bring the observation forms to class and complete the list of assignments in 1.6 on the wiki.

The following were main focuses of today's class:

1. Principles of NCTM

2. Technology and its uses

3. Communication

For the Principles, we each reviewed and discussed each of our googledocs. The 6 principles discussed were equity, curriculum, assessment, technology, learning, and teaching. We brought up questions like how should technology be used in class, how much inequity do we see in schools, how is the curriculum designed and why is it designed in that way, and how do students learn best? We discussed these questions and figured out that much of it depends on the individual teacher's styles and beliefs. We also heard from Susan about an experience in which a calculator would be a very useful tool and it could not be used.

We discussed technology because in the Principles, Nicole and I discussed how we felt that it would be often used as a crutch, and then discussed whether or not we felt it was a good thing to use in the classroom or not, and if there was a time and place for the use of technology, especially calculators. The general consensus is that technology should be used, but the students also need to understand the basics, like multiplication tables, division, addition, and subtraction.

The discussion on technology, along with today's in-class apparatus about representation brought up the topic of communication, and what is the most effective way. This is shown in the representation of 3/4s sheet, because there are many ways to convey the same meaning, and some students respond better to some than they do others. The representation worksheet was looking at different representations and determining what we thought, and this leads to communication because sometimes the teacher and student do not always have the same understanding, possibly because something was misrepresented.

<span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 140%;">Session 5 Jan. 24, 2012: <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">REMINDERS: <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">1. MERPs: Email your revised copies to Dr. B ASAP electronically. <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2. Add to Googledocs "from theory to practice" page observations you made in the middle school classroom last week and how they relate to articles and discussions we have explored. <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">3. "Divide and Conquer"-- Check out the section "Assignments and Reminders" for your group's reading assignment. For these, make sure you pay attention to the main points listed in the margins and support these through the reading. Add examples.

<span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">DISCUSSION POINTS: <span style="color: #ff0000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">(Now that you've written this, I'm wondering if you went through it again and asked yourself "So what big ideas related to teaching comes through?" and then reorganized the summary around those big ideas or themes, how different would your summary look? This is a question for all of you to think about when summarizing the class.) <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">We reviewed and made connections with NCTM Standards and assigned readings. Here is what we explored:

<span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Principles Make sure you made corrections based off of Dr. B's suggestions for these on googledocs <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">a. <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; text-decoration: line-through;">Assignments I think you mean Assessment <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">b. Technology <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">c. Equity <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">d. Learning <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">e. Curriculum <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">f. Teaching/<span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; text-decoration: line-through;"> learning <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Standards: <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">--- Content (what material we need students to know) <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">--- Process (How we will cover this content) <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">We also discussed major points/themes throughout the readings. <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">1. Cooperative learning vs. group work: Many teachers do not implement true cooperative learning, but rather end up with group work-- which is a problem because students do not experience positive interdependence (where they rely on and use one another to experience learning).The Kagan (spelling?) model was used for team building to set the framework for cooperative learning (Dr. B is offering us say in whether we want to take this bunny trail, let her know if you think we should). <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">2. Umbeck discussed "launch-explore-summarize" which she warns will make students uncomfortable, but getting students out of their comfort zones must happen to challenge them in mathematics. <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">3. Frank's argument that students' views and beliefs of mathematics shape how much they learn: The quiz that we were given in class a couple weeks ago came from this article. Frank argues that these beliefs need to be changed for problem solving skills to develop.

<span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">Hats and scarfs SOLVED! <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">We revisited our hats and scarfs problem after having some time on our own to think it through (and we all seemed to value having this time, something we need to keep in mind with our students) and we finally have some answers! We realized that having to "prove" to ourselves that indeed common numerators was the trick ( <span style="color: #ff0000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">I try to avoid the word trick. Perhaps we could say "that is what we needed to notice mathematically"? Other ideas?) <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">to this problem and not finding a common denominator which we have been brainwashed to believe "always works" (and in the right context, we know, it will.. just not when that's not what the problem is asking!). There were several methods offered: both pictorial representations, algebraic solutions, and also this "add-add" method. Exploring similarities in the various methods provided insight into the problem; when Josh and Susan shared their algebraic solutions, it appeared they had quite different approaches, but a closer examination showed they both used substitution and variable isolation in solving the problem. <span style="color: #0f5c5c; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">In addition, the question of how middle school students must feel when presented material without reason supporting it was explored-- we all went "what the heck!" in our heads when the add-add method was introduced just like our students will feel if presented material in this manner. We also felt the benefit of having to figure out and reason by ourselves in discovering how to solve the problem, just as our students will achieve that sense of reward when they make discoveries. Good work, team! <span style="color: #ff0000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif;">(I like that reference to cooperative learning! : ))

Session 4 Thurs Jan 19 At KPS Maple Street Magnet School for the Arts

Session 3 Tuesday Jan 17
 * <span style="font-family: 'Arial','sans-serif';">General Announcements and Reminders **


 * <span style="font-family: 'Arial','sans-serif';">We reviewed the use of our electronic tools for classroom use. Email will be used for announcements and should be checked daily. WikiSpaces is a working space for all students to share. Please check daily. GOOGLE Docs will be used for work that we will complete in class together.


 * <span style="font-family: 'Arial','sans-serif';">We will visit Maple Street Middle School on Thursday, January 19. Please be at the school by 7:20am in professional dress. Bring a writing instrument. Our classroom observation will be from 7:35am to 8:35am. We will meet afterward as a group in room 508 from 8:39am to 9:39am to discuss our observations.


 * <span style="font-family: 'Arial','sans-serif';">Homework: (a) Solve the hats/scarves problem on your own before Thursday. Due next Tuesday in your notebook. (b) Read “Student Confidence and Student Involvement”. Submit a one page reflection on Wiki. The Wiki posting is due Wednesday night before 11:59pm. (c) Read Problem Solving process standards for Grades 6-8 in the Principles and Standards by next Tuesday. Keep notes in your notebook.


 * <span style="font-family: 'Arial','sans-serif';">The following ideas were the focus of today’s class: **

<span style="font-family: 'Arial','sans-serif';">1. Problem Solving Scenarios

<span style="font-family: 'Arial','sans-serif';">2. Mathematical Beliefs Survey

<span style="font-family: 'Arial','sans-serif';">3. The “Egg”

<span style="font-family: 'Arial','sans-serif';">Two different problem solving scenarios (1) were experienced in class with the “hats and scarves” problem. We were initially given a minute to try to solve the problem on our own. Dr. B. then __gave__ us a procedure for solving the problem. The procedure included calculations that were foreign or questionable to many of us (least common numerator, the ADD-ADD method). No explanation was given on how the procedure was derived. It was easy to memorize, but we had no understanding why it worked. We were the “receivers” of information. This is a very traditional teaching model in the mathematics classroom.

<span style="font-family: 'Arial','sans-serif';">After this, Dr. B. let us try to solve the problem with our group. As many of us experienced, we were frustrated and realized that the answer would not come quickly or easily. Did we give up?

<span style="font-family: 'Arial','sans-serif';">What did these two problem solving scenarios demonstrate? The activity challenged us in some of our beliefs as students. We realized that some problems cannot be solved quickly. We may have felt more comfortable in the traditional student role of “receiver” vs. “constructor”. Is “getting the final answer” the only thing that is important when we are gaining mathematical knowledge as a student? How locked are we in traditional teacher/student roles and other mathematical beliefs?

<span style="font-family: 'Arial','sans-serif';">We discussed the results of the mathematical beliefs survey (2) that we took last Thursday. We compared our class results to the cumulative results of some of Dr. B’s prior classes.

<span style="font-family: 'Arial','sans-serif';">How do the beliefs of students affect their expectation about learning? How do their expectations affect the role of teachers? How does a teacher’s belief about learning affect his/her teaching style? How can we shift the belief of the student learner to one of “finder-constructor” instead of “receiver”? Can teachers accept their new role as one who guides and supports learning – not as transmitters of information? Wow, this is some challenging stuff to think about! What should we do?

<span style="font-family: 'Arial','sans-serif';">The survey showed that we had a long way to go in order to change our own thinking as students. Until we change our own beliefs, we will teach the way we were taught. Our students will have the same belief system that we have. The road to mathematical knowledge will not change until teachers change. We need to let our students solve problems on their own.

<span style="font-family: 'Arial','sans-serif';">When discussing our beliefs and how they will affect our teaching, we discussed the “Egg” model (3) for mathematical knowledge for teaching. It is divided into the two main categories of subject matter knowledge and pedagogical knowledge for teaching. Not only do teachers have to know mathematical content (specific, common, horizon), but need to integrate the knowledge of “how” to teach mathematics. Our mathematical belief system, as surveyed in class, dictates our pedagogical practices in the classroom.

Session 2 Thursday, Jan 12
 * ====== First off I would like to remind you that we have three assignments to be completed over the weekend: a reflection paper over the two assigned readings, your part to be added onto the summary of A Vision for School Mathematics that the whole class will work on, and also the Principles for School Mathematics summary that you and your partner(s) were assigned. ======
 * Now, there were a few things we talked about in class. First was all about the standards (which ones?) in schools for middle school students, and how those standards affect teachers. We discussed how difficult it is for teachers to try to instill a way of thinking in our students to help them become independent learners, but also teach them in such a way that they meet the standards set for the students when tested. Is there any hope of accomplishing both?
 * Next, Dr. Browning had us read the sample reflections in the packet handed to us on Tuesday, and we deliberated why one was better than the other . while using the rubric provided . Keep in mind to take it a step further in your reflection papers than just the initial spark of thought!
 * Lastly, we talked about Subject Matter Knowledge vs. maybe and instead of vs . Pedagogical Content Knowledge that was displayed in "the egg". We discussed how teachers need to learn subjects in a different way than other professions, such as engineers. Teachers need to know more of the "why" and engineers need to know more of the "how". Do teachers need to understand some of the "how" to apply mathematics as well?
 * I'm pretty sure I covered most of what we discussed in class. If I left anything out let me know! Have a good weekend everybody :)