The Teachers' Scrounge

News and comments from the world of public education. A middle school math teacher shared what he learned today.

Wednesday, April 30, 2008

It depends what the definition of "is" is.

In the recent issue of Mathematics Teaching in the Middle School, I found an insightful article about the EQUAL SIGN. Who knew?

The authors gave students the following questionnaire:

The following questions are about this statement:
3 + 4 = 7
  • What is the name of the red symbol?
  • What does the symbol mean?
  • Can the symbol mean anything else? If yes, please explain.

I guessed my students would tell me that symbol means, "the same as," or, "both sides of the equation have the same value." Boy, was I wrong!

The article explains that more than half of the middle schoolers surveyed say that the equal sign is "operational" -- that is, they say it means, "find the sum of 3 and 4," or, "the answer to the problem on the left." The equal sign is actually "relational" -- something like, "is the same as," or, "the numbers on both sides are the same amount."

Well, what about my 8th-grade Algebra I students? I gave this questionnaire to 24 of my Algebra students, and roughly two-thirds of my students gave operational answers such as,
  • The symbol shows that the answer is next to it.
  • The two #s before it, when put together by the proper action of the symbol between it, will equal the number coming after it.
  • It means the question should be solved.
  • It means the two numbers sun is something when put together.
WOW. Algebra I students... successful, bright Algebra I students. I persuaded one of my colleagues to survey his pre-Algebra 8th graders, and the percentages were EXACTLY the same. Two-thirds of the students misunderstand the equal sign.

Does this matter? Sure. The article presents great evidence, too. They asked students to identify the value of the variable in problems like, 5m + 2 = 52. Students with a proper [relational] understanding of the equal sign answered these questions correctly TWICE as often as other students. And this makes sense to me. If a student has a poor understanding of the equal sign, problems like the one above are nonsensical. Solving them becomes a dance of gimmicks and rules instead of internalizing relationships. Could correcting such a BASIC piece of information result in better student performance in increased understanding?

We want students to understand their mathematics -- not simply sleepwalk through rote algorithms. It is impossible to achieve this level of algebraic reasoning without understanding the relationship represented by the equal sign.

I'm shocked that I have allowed my students to escape such a basic understanding. It makes me wonder what other basic skills and definitions could REALLY benefit my students.


See this described at Improbable Research

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Tuesday, April 29, 2008

Nobody said anything!

Some of the common "new-teacher mistakes" are pretty well known. For example, trying to be friends with your students (a mistake that too many parents seem to make, as well). In my early days as a teacher, I felt terrible when a student was "hurt" by my correction. I learned my mistake the same way many new teachers have -- Harry and Margaret Wong's book, The First Days of School. This book helped me overcome my fear of correction. I used to rely too much on the "ignore it and it will stop" technique. There are some behaviors that can be discouraged by refusing to give them attention. But many behaviors do not fall in this category.

I am continually amazed by how many times a student is just waiting for someone to tell them to stop. All sorts of activities -- from throwing bouncy balls to wearing low-cut blouses -- kids will [often] pleasantly stop when asked. I knew this was generally true when boys are fighting. They often just need an excuse to stop, and a teacher yelling "stop" will suffice. But I never realized how many other behaviors were the same way. Kids who violate dress code, but bring an appropriate pair of pants and just wait for someone to tell them to change.

So now I usually verbally correct everything. And I embrace the "Love and Logic" technique of expecting compliance. I tell a student what they should be doing ("tuck that shirt in before you get back to class," "pick up the trash around your desk," "stop drawing mustaches on your face") and walk away. With no one to argue with, they often comply. And even if they don't, they can't tell the next teacher that no one has said anything before.

Of course, there is a downside to this. I recently got in trouble picking up the niece and nephew from daycare. I corrected some child who was claiming he had the right to put other kids in a headlock. I told him he was wrong and he knew better. I thought that was the end of it... then his mom told me I should have addressed her over the problem. Maybe I should have referred her to Harry Wong's book.

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Monday, April 28, 2008

What was this test supposed to tell us, anyway?

This is TAKS testing week, so the topic is on everyone's mind. This year there are new rules and regulations... to keep an eye on teachers.

Odd, but it seems lately the State is more concerned with teachers' dishonesty. This is a symptom of another problem. This test is designed to assess the students' knowledge and skills. But the results are being used to assess the teachers, campus, and district. That's not what the test was designed to do. When we use the test for other goals, it become ineffective at both tasks. Not only is the test being used to judge something it was never meant to gauge, but it means that teachers, schools, and districts start doing things the make the test a poor assessment of students' knowledge and skills. There is pressure to teach test-taking, shortcuts, only key ideas, etc.

Use the test for what it was designed to do.


Sunday, April 27, 2008

When will I use this?

What exactly did we sign up for? When I chose to major in mathematics, I had no idea the type of coursework that would entail. Oh, I thought I did. I thought mathematics = arithmetic. Instead I spent time in classes learning about Fourier Series, Fundamental Theorems of this or that, and the origins of irrational numbers. Dr. Gawande (whose book I mentioned previously) writes that he was surprised that the most difficult part of his job is not the medical knowledge, but rather when and how to apply the power of this knowledge. Teachers know the majority of our challenges arise, not from the content, but from classroom management.

In high school I knew I wanted to become a teacher. Early in high school -- like freshman year. I was spending every class period, every day in a perfect setting to learn about my future profession. I was observing classroom teachers for six hours, five days a week. I had effective teachers, poor teachers, traditional classrooms, interactive groups, core curriculum subjects, electives, assertive discipline, no discipline, and more! Despite the wealth that was spread before me, I did not make an effort to tap the wisdom displayed before me daily. I didn't pepper my teachers with questions about how they accomplished their work. I didn't ask how they were able to grade 150 tests overnight, what discipline problems were most challenging, when they preferred to phone parents at home, or what they did when 40 students were failing because of missing assignments.

The most information I received was pushed to me by the teacher who (knowingly or not) recruited me into the profession. Mr. H taught 9th-grade geometry. In a portable building. I soon found myself hanging out in Mr. H's room before school, after school, and sometimes even at lunch -- for FOUR YEARS. In retrospect, I don't know how he could stand me.

Mr. H shared a few inside secrets of his profession. "When I give the first test of the year, I sit in the back of the room. You can spot the cheaters because they keep turning around in their desks to keep an eye on the teacher." I didn't ask for this information, but I think Mr. H knew that these insights would serve me well. So he told me whether I asked or not. "Two female students who I busted for cheating have been trying to stop by for help before school. They stand real close to me and rub up against me. I've gone down to let the principal know and I've told the students we can go to the library when they need homework help."

Sometime before I retire, I hope I recruit a replacement. I hope he or she picks up a few ideas from my classroom that will make his or her classroom run more smoothly. They won't know what they are getting into.

The workforce is changing. Many of my students will hold jobs that have not been invented yet. (Can you imagine if I told my high school counselor that I wanted to be a webmaster or a network manager?) Even if my students know what jobs they will settle on (after the normal career changes), they -- like me -- probably have no clue what skills that job will require. I didn't realize that managing restroom passes and spare pencils would take more effort than teaching slopes and intercepts.

Once upon a time, we teachers would develop answers for that annoying question: "When will we use this?" We exchanged samples of real world applications, we traded stories of math-intensive careers (from airline pilots to nurses), and we suggested pithy replies ("you'll use this Wednesday... on the test!"). Recently we have been laughing at the question. How can I possibly predict how you will use this? I don't even describe the job market and skill sets of the future. But I know that employers want someone who is trainable, teachable. So practice by allowing me to teach you this now.


Saturday, April 26, 2008

Do what you can, then worry about the rest

Margie delivers newspapers to 100 houses. She pays $3.00 for a pack of 25 newspapers. The subscribers pay 50 cents for each newspaper. Margie delivers newspapers 6 days per week. How much profit does she make each week?
One of my colleagues laments that his students freeze up when faced with questions like the one above. He says they sit back in their desks, turn up their noses, and say, "I don't know what to do." There is so much information here! Everyone can do something with this information. Tell me how many packs of newspapers she buys? What is Margie's cost-per-paper? How many papers does she sell in a week? SOMETHING!

Students may not realize that we give them very leading questions. If they calculate what they can, the resulting information takes them to another step, then another, until it has led them down a path to the required solution.

One strategy I want to attempt -- have students write five questions they can answer with the information at the beginning of this post. Help students develop problem solving strategies by writing the problems.

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