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!