*Out in Left Field proudly presents the seventh in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names (with the exception of Miss Katharine's) have been changed to protect privacy.*

**Trying to Hold On and Another Guest on the Raft**

I was thinking of not writing any more letters since it seems no one is reading them. Miss Katharine just about had a heart attack when I told her this. She said a lot of people are reading them and if I am going to quit writing them, why would I leave the readers hanging after telling them my teacher left for a family emergency and I was on my own? So, I’m not quitting. And I'll tell you know that I got through it all just fine. My teacher’s father would pass away and she would be gone for two weeks during which time I was pretty much on my own.

I did get some help the very next day. With five minutes to go before the first period bell, and no substitute, I had given up on anyone showing. But then I heard a key in the door and eagerly went to open it. This caused a problem. “I can’t get the key out,” the sub said. “Close the door.” I did so; he got the key out and opened the door. “My name is Jaime Ortega." he said. "Sorry I’m late; there was traffic. Is there a seating chart?”

I handed him the chart. "Yes," he said like a doctor looking at an X-ray. “I know a lot of these kids,” he told me. "Some of them I know from when I subbed at the elementary schools. And I know some of their brother and sisters.” He was a former student teacher of Tina’s as well, and had subbed in the school many times. He was also from the area. “I’m hungry,” he said. “I worked out this morning and didn’t get a chance to eat breakfast. We get a break after first period?” he asked.

For the next two weeks our routine began with Jaime telling me he was hungry and taking attendance. I would check in homework. After that I would do the warm-up problems, go over yesterday’s homework, and start the day’s lesson. Jaime would help answer students' questions and work with students who had trouble focusing.

That first day, we continued to work with exponents—this time presenting the rules for multiplying and dividing powers of the same base, like 5

^{6}/5

^{2}= 5

^{4}. Which actually should have been the lesson for the day before so as to better explain negative and zero exponents.

After first period would come a break, and Jaime's daily routine was to go to the teacher's lounge and eat whatever snack he happened to have on hand.

Second period was the discovery-based algebra class. I checked in homework, Jaime would go over the answers to the homework; then I would start the day's lesson. It was about functions and graphing. From the second week of class they were “finding the pattern” of growth in a variety of problems. For example, a person buys a tree seedling that is 3 feet tall and plants it. It then grows at a rate of 2 feet per month. Students are asked how tall the tree is after 2 months, 5 months, and finally x months. They are supposed to figure out that 3 feet is the starting point, so they will end up with the equation y = 2x + 3 which they then graph and notice that the starting point of 3 crosses the y axis.

CPM wants students, to "make connections” between the patterns of growth, the equation for such pattern, table of values, graph of the equation and word description of the patterns. After all this, the book finally provides a definition of y intercept and slope and the standard form of the equation of the line (y = mx + b). And then students get to “put it all together”. Except for figuring out slope from any two points; that was still three chapters away.

The authors believe that “simply memorizing what to do in a specific situation without an understanding of the reasons why the method works too often leads to quick forgetting and no real long-term learning.” To rectify this sad situation, students are “asked to solve problems designed to develop the method.” In other words, teaching procedures directly never leads to understanding--or connections.

Rudy was a very bright student, and despite earlier confusions in which he referred to 2x + 5 as “paste," he was definitely the “go to” person within his group of four. His group liked me because of my unusual habit of answering questions. He asked me about a problem that asked for the equation of the graph of a line. “Do you know what the y intercept is?” I asked.

“Where it crosses the y axis?” he asked.

“Yes. Do you know the equation y = mx + b?” I asked pointing to the equation I had written on the board earlier. He nodded. “So what is the y intercept in that equation?”

I reminded him of his past "connections": How to plug in the intercept value for ‘b’, how to compute slope, how ‘m’ means ‘slope’ in that equation. His eyes got big and he began speaking in Spanish to his group. Had Tina been there she would have told me I was giving them the answer. And I was—sort of. The problem-based "connections" approach wasn’t leading to "aha" moments for many students. I noticed Jaime giving similarly explicit answers to the other groups.

Later in the day during 4th period prep, I went to the teacher’s lounge. Jaime was there, eating salsa chips that he sprinked with Tapatio sauce.

“I thought you had left for the day,” I said.

“I was hungry,” he said. “Chip?”

“Too hot for me,” I said.

He appeared to be thinking about something. “You know, I am tutoring some students in calculus at the high school here and they use the CPM book for that. I don't think it’s very good. You have to be really smart to see what's going on.”

“I’m with you.”

“Any word on when Tina is coming back?” he asked. I said no one knew anything. In the meantime, though, it was nice to have somebody with me on the raft--someone who was not afraid to say the sun was not rising in the south.

## 9 comments:

I am mostly a lurker, but I definitely read these -- don't stop!

The boots on the ground perspective is very, very valuable.

"The problem-based "connections" approach wasn’t leading to "aha" moments for many students."

Most all of my direct instruction teachers - through 7 1/2 years of college, tried to lead us students to light bulb moments. The funny ones were when they kept giving clues until they basically had to tell us the answer. Then, of course, only a couple might "get it". What about the rest? They were directly given the answer. The same goes for student-centered groups, except that it's always one student directly teaching the rest. So much for the "aha" moment for them. In general, few ever get true "aha" moments, and those happen only rarely. A learning environment that is centered on discovery is extraordinarily slow and inefficient. It will, however, warm the cockles of any well-indoctrinated teacher. Hey! Do ed schools use discovery learning? Or, is it kind of like leading the witness?

Huck - I love your articles - they're one of my favorites to read. So, if you suddenly decide not to continue the series, I will personally form a posse to hunt you down and force you to draw squirrels to demonstrate your fluency with basic arithmetic for the rest of your life. As my 8 year old often reminds me, squirrels are not triangles - they have have 4 legs, pointed noses, whiskers, and a big fuzzy tail.

Be warned Huck, be warned. The squirrel posse is waiting........

I'm another fan. I like your stories and your thoughtful perspective. I hope you can stand to keep writing.

Amen. Keep writing.

I've been reading, too, just not leaving comments. I've experimented with discovery-based approaches as a math teacher and have pretty much abandoned them. I do use hands-on stuff, but only in a very careful, efficient, and explicit way with a clear point. The real "aha" moment for me was when my own son (who is mathematically gifted with an IQ in the 140s) took a precalculus class with the discovery approach at his highly regarded boarding school. He hated it, had many Cs and struggled to learn. The next year he had Calculus AB with a teacher who used direct instruction- making great connections and application- and he was back to As. If it doesn't even work for the gifted kids.... what in the world are we doing?

It has long been my observation that those on the left/progressive end of the spectrum are less interested in the actual outcomes/results of their interventions than they are in the nobility/intensity of their intentions. It's all about their feelings; unintended consequences are irrelevant.

SteveH asked if ed schools use discovery learning. Indeed, I went to one that did. We were given a problem (it had to do with tessellations) and were given two weeks to discover whatever it was that we were supposed to discover. I was the oldest one in the class, with small children at home, so that my time was limited. After two weeks, I was the only one who "discovered" what we were supposed to discover. I was quite unhappy about it because, had someone explained the concept to me, I'd have had it in a minute or less and would have easily been able to apply it. I learned two things: Preservice teachers (at least those in my class) do not want to discover, don't know what to discover, and don't know how to discover it. This leads to the question, why would they want their students to do what they were unable/unwilling to do? Secondly, I learned that I didn't like it, that it was inefficient, and that as a learner, it frustrated me and made me angry -- enough that it turned me off to the subject matter.

Huck, you're doing a great job, and keep those letters coming!

I think why I have such a negative reaction to constructivism is that it seems to me the educational equivalent of that dreadful relationship tactic of "Something is wrong, but I'm not going to tell you what it is, you have to figure it out for yourself."

I recall a first or second season Monk episode used that manipulation technique -- and framed it as if it were a *good* thing -- and I wanted to throw a book at the TV.

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