Logistics

How do teachers keep all the papers straight from all their different classes?  This is something I'm learning from my CT.  One thing I've picked up that I really like is that she has laminated folders, one for each period we teach.  Inside the folder is grade sheets for particular assignments, such as mastery tests, and clean copies of what worksheets we are using that day.  When a student is absent, she writes their name on the worksheet and attaches it to the outside of that period's folder with a paper clip.  Due to our block period, it is hard to always remember who was absent the last time we saw this class.  When the student returns, we can easily see what worksheets they missed and give them to him or her.

"I can" sheets

Our classes are using "I can" sheets: these are filled out at the beginning of the unit and list the important standards for the unit (eg, I can represent a function in an equation, table, words, or graph).  Students self-evaluate themselves on how well they can do each standard on a scale of 1 (I don't know what this means) to 5 (I know this so well I could teach this to the class).  At the beginning of the unit, most of the standards are closer to 1's and 2's.  However, in algebra, many of the students put down 4 or 5 for "I can recognize independent and dependent variables."  Apparently, science classes have been stressing that, and it showed up in math.  After teaching for 2/3 of the unit, we gave the "I can" sheets back to students, asking them to fill them out again for the same standards.  Most of the students now have 4's and 5's.  What I like about the "I can" sheets is the ability to quickly scan the pages and identify students who are still unclear on important concepts.  This enables me to approach them individually and encourage them to come in afterschool or during tutorial in order to help them succeed.  If several students are still unclear on one particular standard, I can reteach that concept in the next lesson.

Simplifying fractions

I got to teach all 5 classes yesterday because my CT was at a training; her substitute was helpful, providing me feedback and ideas about teaching math.  He reminded me how we learned to simplify fractions, by writing 15/20 as 5*3/5*4 and then canceling the 5/5.  This was something so ingrained in my thinking that I didn't usually show that step, yet many of our students have trouble with that step.  Usually, when I'm teaching, I'll ask the students if a fraction can be simplified, but I wasn't scaffolding the process with the intermediate step.  Now I'll write that step down explicitly, so it's an instant review for students who have trouble simplifying fractions.

Substitutes

I know that substitutes are necessary to keep the schools running, but what I don't understand is the lax requirements.  Case in point: our sub today wasn't sure how to evaluate -8 - (-4) because she hadn't done subtracting a negative number lately.  While I understand that is not something one does every day, she is a fully certified teacher for K-8, and she subs in preschool through high school.  How do we expect our 7th graders to learn the basics in math if their teachers aren't comfortable with the basics in math?  If I become a high school math teacher, I'm going to find a sub that I can trust - or else have a foolproof plan for when I get sick.  Egads.

Two-way communication

Today I saw another great stride for D.  Last week, after meeting with him and his mother, I had seen an improvement in work ethic.  When we assigned 3 figures to graph (testing for similarity), he chose to graph all 5 figures, which must have taken a bit of extra time because the coordinates needed to be calculated for each figure.  Today when we taught mental tricks to calculate 1%, 2%, 5%, 10%, 25%, and 50% of a number, I handed out a worksheet for the students to do.  Unfortunately, their skills were not as developed as I thought, and the problems were taking more time than I had planned for the students to complete.  I changed the requirement to finishing 3 multi-step problems, and then students could transition to Math Whizz, an online math program. D gave me his worksheet, and then asked if he could have it back to work on the additional problems.  He said "I want to do more of these."  This from someone who doesn't view himself as able to do math. I can't wait to email his mom and tell her what a terrific attitude her son has!

What's in your hip pocket?

Today I showed up at school and found out that my cooperating teacher was staying home sick and a substitute was taking her place.  Since I was more familiar with the material, I taught the classes today and the substitute helped with grading quizzes, etc.  My cooperating teacher had written the outline for today's classes on the board yesterday, and we had gone over the plans together yesterday, so I felt prepared for the classes.  Except for one detail: our 4th period class was supposed to start with a puzzle, and I didn't know what that puzzle was.  I searched her notebook where she keeps papers, looked through the folders on her table where she organizes the papers for the day, and still no puzzle appeared.  After searching for many minutes, I emailed her and asked where I could find the puzzle, and she responded by saying she didn't have anything specific, but planned to come up with something.  Oh, I can do that!  We did 2 quick puzzles out of Michael Serra's Mathercise, one on directions (N,S,E,W) and one on reading a sentence correctly.  I thought they would be too easy, but several students struggled with the ideas.  I think we need to do more of them!

Change in performance

I was thrilled today when D brought in his homework, and he had done above and beyond what we had asked of the class.  There were 5 figures to graph to evaluate for similarity, and we had divided the class into two groups: one group did the baseline and two distortions, the other group did the baseline and two different distortions.  Within each group, they should have found 2 similar figures and one impostor.  Most of the class had excuses about why they hadn't done it, and we ended up giving them class time this morning to finish their homework.  Imagine my surprise when D shows us all 5 figures completed.  I publicly acknowledged his hard work and my CT actually gave out candy for students who did the homework at home.  I hope this signals a new work ethic.  Perhaps last week's conference with D and his mom made a difference.  His attitude certainly made my day!

Two ideas that struck me

From chapter 9 in our methods book, I found two ideas that really struck me as worthwhile.  One pertaining to gender equity said that, in general, girls are less comfortable with calculators and technology than boys.  I remember during my freshman year of college when I started my engineering classes that all of the boys were eager to program on computers and I had no experience with any programming.  Not only did I feel intimidated by the boys already having the knowledge, but I felt that somehow I had missed something in high school that I was supposed to learn, and it made me reluctant to ask for help.  So this tidbit of information really struck home with me.  Today during a practice session with linear functions, I encouraged the students to use our graphing calculators.  Two girls chose to use them, so I felt like I made a step in the right direction.

The other idea I liked was to find mathematical problems from different cultures to share with my students. With our diverse student population ranging from Vietnamese to Ukrainian to Gabonese, I'm eager to search for useful math ideas that will shed light on their various cultures.  I think it will be healthy for all students to see the ways math is used in different situations.

Catching up is hard to do

Students who have realized that the end of the trimester is only a few days away are now bringing in missed assignments they think will bring their grade up.  However, our standards-based grading only counts tests and quizzes for grades, so showing us their homework doesn't affect their grades.  Other students are asking to retake tests, so we have set up times before and after school, as well as during lunch tutorial, for students to come and take a different version of their unit test.  It seems odd to me that students who go to the trouble of taking another test have not made an effort to study the questions they missed on the first test. In some cases, the students miss questions that are very similar to ones they got right the first time.  Or they'll get the same questions wrong that they missed the first time (not every question is different).  I wonder if it's due to parental motivation that the students are retaking tests; it hasn't brought up anyone's grade so far.  We had one student planning to retake the test tomorrow, but now school is cancelled due to snow.  There's always next trimester!

It doesn’t count if it doesn’t stick

I saw this comment today when we had snow predicted (and school got out two hours early), but when I first read it, I thought about how appropriate it is for teaching.  In teaching to the students, it doesn’t count as valid teaching if the concepts don’t stick with our students.  We are working on how to make similar figures in 7th grade math, and I’m stressing that similar figures need to have the same scale factor for each dimension.  I keep asking “what number do you need to multiply by to go from the old figure to the new figure?” and defining this as the scale factor.  We did multiple examples with both rectangles and triangles, and demonstrated the scale factor is bigger than 1 when we are going from a small shape to a large shape, and the reciprocal going from the big shape to the smaller.  At the end of the class, I gave the students an exit ticket with two sets of triangles: one set was similar shapes because each side was multiplied by 2 to get the other side, and the other set had triangles whose dimensions were changed by an addition of 1 to each side.  Five students out of 24 got it wrong, so at least the majority were following along.  But for those five, why did they think those triangles were similar?  The right idea didn’t stick.

Low point

So I'm gone from home more than 12 hours straight due to teaching school and going to school, I haven't seen my husband today since he was gone at a meeting by the time I got home, I've had an hour-long fight with my son about one thing after another, there's two loads of laundry to fold and I'm wondering for the umpteenth time: what did I get myself into?  I have a lesson plan to write for tomorrow, another one I need to change based on what happened today, a unit plan to write for methods and I'm beat.  Why did I want to do this?  This is too much!  I hope the sun shines tomorrow because I need it!

Finding mistakes in my teaching

My cooperating teacher and I have been co-planning our classes together, and I take her plans and flesh them out into more formal plans for methods class. However, I'm only doing that for the Algebra and CMP2 classes, not for our Study Tech class.  So when I started teaching Study Tech today, I realized that I didn't have a detailed feeling of where our class needed to go.  Frequently, we are previewing or reviewing concepts these students will see in their CMP2 class.  We were working on equivalent fractions today, something all of the students struggle with to some degree.  I reviewed the worksheet that they had for homework by asking students how they solved the problems, and then we gave the students another worksheet to practice with positive and negative numbers.  After 5 minutes of working with the students, I saw they needed more scaffolding, so I asked them to write down each time the fraction they multiplied by to get the equivalent fraction (ie, 5/5, 3/3, 8/8, etc.) Most students kept working on the problems, but in going around to assess their progress, none of them were writing down the intermediate step.  I didn't make that step mandatory, so they weren't doing it, even though it would have helped most students understand the process. I wish I could go back and reteach this lesson, making sure that students write down the ratio they are using to find their answer.  Next time, I'll know!

Start with the basics

We gave our Study Tech students an online assessment that was targeted to material taught in the 6th-8th grades, and many of them did poorly on the test.  These are students who got a 1 on the MSP test at the end of 6th grade.  Because we want to clearly define where they are now in order to show improvement later, we also gave these students the online assessment for grades K-5.  Several of our students scored at the 3rd or 4th grade level in certain subjects, particularly working with fractions and finding equivalent fractions.  This topic is going to come up again in later math, so we really want to make sure they understand this basic idea.  Our realization that we need to go back to concepts they should have learned in 3rd grade opened our eyes to the level of basic math skills these kids are missing.

What is your math identity?

Last Tuesday's methods class touched on the concept of math identities: what is your relationship to math?  How do you see yourself - good at math or not? Does the idea of solving math problems bring you peace or anxiety?  The article we read really got me thinking about our Study Tech kids, students who struggle at math and have been labeled as "not meeting standard."  Every day, I aim to make math relevant for these students because they clearly believe that math is not important to their future. If we can show them how math is relevant, then I think they may be more self-motivated about their learning.  In particular, there is one student who I'm tutoring who is bored or sleepy during every class. He can do the work, but has trouble getting started.  We gave students a sudoku puzzle to do after finishing a quiz because they needed to stay quiet as the other students worked on their quiz.  I was surprised and delighted when I handed my tutored student a sudoku, and he said "Oh, I love these. I have a book of them at home."  Now I have a hook to work with! I want to give these students a chance to see themselves at mathematicians, so they will have more self-confidence to tackle math problems.

Optimal planning

Now on the fourth iteration of my plan for this Friday's observation, I keep coming back to what I'm going to present and what the students are going to do, how they are going to learn, what do I scaffold and what do I leave for them to struggle with.  At first, I had the straightforward, direct-instruction lesson with times for students to work out the problem and I would randomly choose students to answer.  Then I added an entire scaffold on the relationship of a container's volume to height, including borrowing a graduated cylinder from science and comparing data points from the cylinder with a two-cup measure. There's a problem in the book that asks students to match a container shape with a graph, and I saw this as a great mini-lesson in spatial reasoning, which I know from the research may come more easily to boys than girls.  Then I added think-pair-share to the questions, which would give the students an opportunity to interact and get some informal feedback from each other.  And I copied some worksheets which students could do in class and then hand in, allowing me to provide (non-graded) feedback to them for the following Tuesday.

Then I talked it over with my CT and realized that our main focus on Friday is to review for Monday's test, and I need to cut back my plan to keep the new material to 20 minutes or less.  Back to the drawing board!  The worksheet will be used for after the test on Monday, I'll pull some relevant questions from the book for Friday, I'll rewrite the plan and run it past my CT again tomorrow....  Where's another hour of sleep for tonight?

Scaffolding vs. telling

As I'm working with W on adding and subtracting integers, I feel good because I like helping. I try to ask questions to help her understand her strategy, and not give her the answers.  For instance: "How do you subtract a negative?"  I realized yesterday that I have probably helped her too much because as I went around the room checking students' understanding on a review worksheet, she grabbed my wrist and "no, you can't leave me."  Oops! 

I have told the students during instruction that they need to find the best learning tool for them and work with that.  We have shown them several different analogies for working with positive and negative numbers, and given them time to practice with black & red chips, hot & cold cubes, black & red cards, gaining and losing yards in a football game, but it still seems that students will see an equation and jump to an answer without thinking about what it means.  I don't want to tell them the answer, but I'm confused on how to scaffold the learning process.  After we taught a lesson that a negative times a negative equals a positive, I asked them to solve -3 + -8 (with emphasis on the +) and had several students enthusiastically respond +11!

Switching gears

Today I began a lesson in our algebra class on solving absolute value inequalities with a straightforward visualization: a skateboarder going a constant 20 ft/s (really? who makes these numbers up?) is coming towards a bystander who is 100 ft away.  When will the skateboarder be 60 ft away from the bystander (who is not moving)?  The class came up with the concept that there are TWO times the skateboarder is 60 ft away, one on each side.  I compared the bystander to Zero on the number line and explained that the absolute value of a number is how far it is from zero.  I wrote absolute value on the board with a definition, and proceeded to show an equation (absolute value of x = 5).  Several students stopped me to say "what is absolute value?"  It totally threw me for a loop because in my mind, I was already headed towards the inequality part of the lesson, and here the students were telling me they didn't know what absolute value was.  I spent much longer than I intended on the basics (since if they don't get those, the rest is useless), and we didn't even START the group quiz that was going to take the last 45 minutes.  Argh!  We plan carefully, but not carefully enough.  It makes me wonder what are they teaching in elementary school?

A day of tricks

It can be hard to capture the attention of kids the day after Halloween with the world of math. We had wild kids in all of our classes today, and our intentions for math-centered instruction was diverted by discipline issues at times.  Also, I was preparing for my observation on Wednesday, so I spent the planning period incorporating my teacher's feedback.  However, as I went to save the file, I saw an erroneous message that my hard disk was full.  I spent my lunch trying various methods to save the file so that I could send a copy to my clinical faculty.  I was not able to save the file, so I printed a copy, restarted the computer, and spent an hour this evening redoing my revisions.  In addition, I tried to print a color copy at school today, only to discover that the color printer was not working (although it looked fine in the print queue).  All this drama drove home the fact that it's important to plan ahead of time!

Teaching to the plan

I've had the opportunity to teach several classes now.  Frequently, my cooperating teacher will share what she had in mind for the class period and ask me if I'd like to teach.  I enjoy working with students and sharing analogies that I think will help them, so I usually jump at the chance to teach.  The problem is, what she has in mind does not usually get completely transferred to my brain before I start to teach the class.  With algebra, the textbook is more traditional and I'm comfortable with all of the ideas and the overall learning target for each day.  However, with CMP2, I feel that the text does not adequately cover the main ideas and we need to supplement with additional information.  Twice, I have taught the CMP2 class and discovered that I skipped an activity that my cooperating teacher had in mind, but was not clear to me.  I'm learning first-hand that it's important to thoroughly plan beforehand!

Learning by doing

I had the opportunity to teach two of the same class yesterday, back to back.  The subjects were adding and subtracting integers and introducing coordinate graphing.  During the first block period, I addressed questions from the homework, and gave the students questions from the book to do on their own.  The point was to recognize that there are many ways to say the same equation: 12 - 12 is the same as 12 + -12 and -12  - -12 and -12 + 12 all equal zero, but -12 + -12 is -24 and not the same thing.  The students had many questions throughout the period, and we ended up only doing 3 of the 6 activities we had planned for the period.  I did introduce coordinate graphing, attempting to model by drawing a graph free-hand on the white board; however, when I went around the room to see what kids were drawing, I saw several that did not have negative numbers on their axes, and I had forgotten to stress that the axes cross at (0,0). 

During the second period, I managed to stay on time better by not having as many students come to the board to answer the questions, but just having them answer from their seats.  I also decided to model the graphing under the doc cam, where I used a ruler and had the same graph paper the students had.  I stressed that the axes cross at the origin, located at (0,0) and that each axis is just a number line, with both positive and negative numbers.  We had significantly more time at the end of the period to introduce the graphing homework, and were able to check in with students individually to see if they understood the concepts before the period ended.  The second time around was so much better!

Accountable students do their work

I taught 6th period algebra today after watching and taking notes while my cooperating teacher taught 1st period algebra.  Our first item of the period was to review the homework and have students check their answers.  I used the doc cam to post the answers as I asked the students to get out their homework.  I heard "we had homework?" "what was the homework?" "I didn't know there was homework."

Our school is implementing standard-based grading, which doesn't assign points to completing the homework.  Our students know that their homework does not affect their algebra grade, so they choose not to do it.  The theory is that students will be able to self-regulate and only do the homework when they need the practice, but in reality, the majority of the students don't do any of the homework.  It made the first few minutes awkward for me, as I don't want to waste time in class, but I wanted to give the few students who had done the homework a chance to review their answers. I was tempted to call on students, chosen randomly, to show their answers at the board, but I didn't want to spend a lot of time on the homework if students were going to flounder at the board.

Afterwards, my teacher and I discussed that middle-school students are not necessarily mature enough to do work that doesn't get checked. We may institute a homework check that we record with either a missing, incomplete, or complete.  This method would also give parents visibility if students aren't completing their homework.

A new strategy for adding and subtracting integers

We have an intervention class that only has 16 students in it; these students did not meet standard on the 6th-grade state test last year, so all of them struggle with math.  While working with these seventh-grade students, I noticed that they frequently confuse adding and subtracting integers when both positive and negative numbers are involved. When I posed 5 + -7 =?,  students answered 12, -12, 2 and -2.  When I asked students for their strategy, one student said that he thought of -7 as -5 + -2, and then cancelled 5 and -5 since he knew that every number added to its opposite is zero, so the answer was -2.  This strategy helped other students to solve additional problems and decide if the answer should be positive or negative. My teacher and I decided to show this technique to our seventh grade math classes on the following day, so that kids that get confused with adding positives and negatives would have an additional method to solve those problems, as well as a check on whether their overall answer should be positive or negative.

Zip my lips

My most enlightening moment today was feedback from the principal after she had observed my teacher lead a math intervention class.  She told me that I was helping the students by further explaining concepts that my teacher wanted the students to struggle with.  I fully support letting students struggle, and I was surprised that I had given them answers.  I don't want to give answers, I want to provide questions.  It was a great wake-up call from the principal for me to not only zip my lip, but take more notice of how my teacher is encouraging students to do their own thinking.  Once again, I need to be reminded to never say anything a kid can say!

Show me the data

Today I observed a Modeling Math class; this course is meant for students who have completed algebra and geometry, but are not ready for algebra 2.  The textbook is based on investigations and simulations, and students spend a significant portion of their time manipulating data.  The teacher intends for all work to be done in class, unless students are not productive during class, and then they have to finish on their own.  I like that the class started with a real-world problem: bones were discovered that might have belonged to Amelia Earhart.  “Given the lengths of the bones, can we determine if they were hers?”  After discussing the information, it became clear that additional information was needed, such as “what is the relationship between specific bone lengths and overall height?”  The class began an investigation by measuring their own head length and overall height, and recording their group data.  During the next class, they will graph their class data and determine if there is a correlation. 

I am intrigued by this course, and the focus on concrete data.  The teacher said that some students who are successful in her class have never received an A or B in math class, and in her class, they can earn good grades.  I saw one student openly celebrating his high test score “I got a 47 (out of 50): that rocks!”  This class can boost the confidence of students who have struggled in math.  In addition, the material gives the students a solid grasp of data analysis, which then can help them better understand the concepts of algebra 2.  It makes me want to insert concrete data into all of our math classes to help students learn.  

Another instance of relevant data was demonstrated by a teacher who graphed Test Scores as a function of Homework Grades.  No surprise: there was a strong positive correlation between students who did all their homework and high test grades.  But showing the students these data may give them food for thought when they are considering whether they have time to do their math homework on any given night. 

Who does the writing?

One high school math teacher I observed gave out typed notes on a new topic.  One advantage of this is that everyone gets correct information, and students who may have difficulty seeing the whiteboard have notes right in front of them.  A disadvantage of handing out prewritten notes is that the students are not writing the notes; sometimes the act of writing the notes helps the student learn the material, so the teacher takes away the learning opportunity when they give out their notes.  Also, the student doesn’t write down the information in their own words, so if they have language difficulties, another person’s words may make it harder to understand the concept. When a worksheet is given out, it may not make it into the binder, whereas if notes are always written into a math notebook, there’s a single place for reference notes. As a future teacher, I'm wondering if it's better to always have students write their own notes, or if there are times when I'll decide to hand out "formal" notes.

Keeping students accountable

One thing I've noticed in high school is that many teachers provide a warm up for students to work on, so that the teacher can get the class focused on the math topic of the day.  Once the students have been given 5 minutes to work on it, teachers will review the answers.  Some teachers work all the problems and ask if there are any questions.  Other teachers have the students provide the answers, and then ask the class if there are any questions. From what I've observed, if the teacher is going to provide the answers, the students don't usually do the problems.  They wait until the teacher writes the answers down and then copy them.  However, the teachers that choose to have students answer the questions and call on them by name will keep students accountable to the warm up.  I expect these teachers probably also have a better understanding of what content is confusing to their students.

Choose your TAs wisely

In one class that I've observed, the female student TA positioned herself near a couple of boys and spent the entire period chatting with them.  I would tell them to be quiet during the teacher's direct instruction (which took most of the period), but they rarely stopped talking for more than a minute.  (Perhaps the fact that I'm only here for a short time meant that I had no authority in her eyes.)  The TA informed me that she was in Running Start, so she goes to college in the morning before she comes to the high school, and then she works 20 hours/week after school.  It surprised me that someone with that much motivation would purposely disrupt the class.  I guess I saw it as disruptive, but maybe the TA didn't.  I don't know if teachers can choose their TAs, but if you can, make sure it's someone who will add to the class environment, rather than detract.

Good tip for continuity from year to year

One teacher I observed today has students summarize their notes for a chapter test in a specific notebook.  The teacher provides composition books for each student (with their name labeled on the spine), and the notebooks usually stay at school, but students are allowed to take them home on the night before a test. The students can take one to two pages of notes to summarize the main ideas from the chapter, as well as diagrams or formulas.  Tomorrow, the students can use the notebook during the last 10-15 minutes of the test.  At the end of the school year, the students have a notebook that contains all the major ideas from the course, and they can take that with them for the following year of math, building continuity of math learning.  Have you ever forgotten the unit circle values for sine and cosine?  Flip back to chapter 2 of pre-calc, and you'll have all the important points right there.  The other advantage of the notebook is the process of making the summary is a great tool for studying.

A warning in high school

I helped out in a math class for seniors who need to pass this class to graduate.  It's called EMP, short for Evidence of Math Proficiency (I think).  It used to be that the students would learn all year, and then try to pass a single test at the end of the year, but there wasn't a great success rate for that.  Last year, they tried putting together a math portfolio with work throughout the year, and they had a 94% success rate.  Unfortunately, the state decided it cost too much to evaluate the portfolios, so now the students have tests throughout the year, but the test directly follows the teaching of the material.  Yesterday's lesson was on scientific notation and Mrs. G led the students through the methodology of converting numbers into and out of scientific notation.  I was helping some of the students during the working time, just verifying they understood the concepts. 

After class, I asked Mrs. G if I could come back later in the day to help with her 2nd EMP class.  She said "yes, but don't look at the work or offer help to the students in these 2 left rows, as they may become violent; just work with students on the right side of the class."  I hadn't thought offering help could provoke such a strong reaction - now I know to be careful!

Observations on Learning

The students I have been observing are in our regular 7th grade math class, as well as an extra math class for students who have not met standard in 6th grade (called Study Tech). I can tell W. is learning when she voluntarily answers questions, provides an alternative solution, explains a strategy to a friend, holds up her hand to show she understands a concept, and writes review notes in her notebook, color-coded to show different ideas.  I saw her having difficulty learning when she was turning around to chat with friends, when she showed an inability to focus on math, and when she came back to a previously-worked problem and couldn’t explain her steps.  I observed D. sharing a factor tree on the board, and working through a fraction multiplication problem.  He was not learning when he put his head down on the table, did not bring in his homework, was joking with friends, and when he could not complete the quiz.  Much of the difficulty for this student seems to be his disorganization.  He comes to class with a binder full of papers, which are not organized by subject or have a discernible order.

In the Study Tech class, I watched the students during an activity where each pair of two students had rods made from blocks to help understand fractions: some rods were color-coded to show halves, some thirds, some quarters, etc.  Each pair of students held up their rod when we asked for specific fractions.  This activity was designed to help students understand common denominators.  I watched as they tried to add one third and one quarter with the blocks, and the students responded with blank looks.  After a few attempts to add different fractions, we had the students put the blocks away because they didn’t seem to be helping, just confusing the students.  This was an example of a hands-on, interactive activity that I expected would be helpful for students uncomfortable with fractions, but in reality, the students seemed confused by the blocks.

On another day, the teacher was discussing how to divide fractions, which should be a review topic for the seventh graders.  I expected the most difficult part for the students to be remembering that dividing is the same as multiplying the reciprocal.  However, during the work session after the instruction, I checked in with several students and the main misunderstanding came from converting a mixed number to an improper fraction.  The parts that I believe are going to be hard may not match what the students believe is hard.

All of the teachers that I have observed genuinely care for their students, and this seems the most important characteristic for a teacher to have.

Different methods helps learning

I like that my middle school uses block periods for four days each week because I know several districts who are transitioning to block periods; however, it can be a challenge to motivate students for 105 minutes of math.  My cooperating teacher is experienced with block period scheduling, and she uses many different techniques throughout the period.  There are times when students work quietly on their own, completing a worksheet or problems from the book.  At other times, the students have white boards to show their answer to a problem at the front, and student are allowed to work with partners.  One student who was reluctant to fill in a worksheet became very motivated when we switched to an online game, even though the math task (rounding decimals) was the same.

Another technique that has helped us assess whether the students understood place value or estimating fractions has been to hand out notecards with different numbers on them, one to each student.  Then the students self-order themselves into a line, based on their understanding of whether 15/7 is greater or less than the square root of 3 in the case of the algebra students.  This gives the students a chance to get out of their seats and work with their classmates to determine relative values.

The students have also come to the board to demonstrate their factor trees, which allows students to see each other as teachers. The teacher usually asks the class if there are any other ways to find the solution, so students can see several methods all create the correct answer.

Starting school with *students*

I started student teaching last week; it's exciting to have students in the classroom after all the training we've been doing with other staff.  Due to all the interventions that my middle school puts in place, we've been mainly focused on procedural stuff in the beginning and we haven't been able to do math yet.  We did a group ball toss activity to learn names: the students enjoyed it, and it helped me remember most of the students' names on the second day. 

In the seventh grade classes, we had groups of 4 students work on a puzzle that spelled TEAM.  It was not obvious that the puzzle pieces spelled a word, so several groups tried to fit all the pieces into one shape.  Only one group out of each class got the word spelled out in the 10 minutes that we gave them to work on it.  We used this opportunity to discuss what works well to solve problems as a group: everyone helping, trying different strategies, drawing in each person in the group.  My teacher gave the students guidelines about how to work in a group, and showed a powerpoint that detailed conversation tips for groupwork.  One ground rule that she stipulated that I would not have automatically thought of was that every person in the group needs to agree on their question before they ask the teacher for guidance. This encourages the students to talk among themselves before turning to outside sources for assistance.  I noticed a couple of groups that did not include everyone, but the majority of students seem to have experience working within groups and shared the process.

Cool math ideas

I had the privilege of attending a training for high school math teachers last week, led by Kristine Lindeblad. She's terrific; if you get the chance to participate in a training by her, by all means, go for it!

She started by giving us a survey, which we filled out, crumpled into balls and threw around the room to insure anonymity. Then we created human bar graphs to show the results, and whoever had the largest shoe size, most number of siblings, born the furthest away got to report out. We then created two different graphs to show two of the questions, and put TAILS on each of them (Title, Axes, Increments, Labels, Scales). Kristine staged which graphs were shown to the class, and asked us to provide an affirmation and a probing question for each graph. I liked that the exercise built community, gave us an idea of what areas were most important for these teachers, and modeled affirmations and questions, which is a great way to give positive feedback.

We did several other activities throughout the day, and at each step, Kristine kept the focus on our learning, modeling what a student-centered classroom looks like. She asked good questions, but always kept the atmosphere positive and encouraging. She had us read parts of Never Say Anything a Kid Can Say! by Stephen Rinehart, which is a great reminder for me: I need to ask more than I explain. One tip that I will take to heart is "Don't carry a pencil" because then you can't write out the solution for the student; he or she needs to do the writing (and learning!) themselves.  I can't wait to try some of these ideas with the students.