January 5, 1998

Today I tutored both Lara (grade 9-geometry) and her sister Nadia (grade 7-basic math). Lara was working on triangle symmetry-learning the AA postulate and the SSS and SAS similarity theorems. One question she asked that I couldn’t answer (I’ve been taught this a million times, and it has never sunk in) was what the difference is between a theorem and a postulate. I also couldn’t explain why she needed to identify rules with the correct label of a postulate or theorem.

To me, it seems that the way they learn geometry, they’re just supposed to accept things as rules. If the difference between a postulate and a theorem is the amount it’s been proven or the fact that it can’t be disproven or whatever, it makes no difference to them because they just need to know how to apply it. Also, I had a really hard time talking about triangle similarity. I totally knew what I meant to say, but I found it to be confusing to explain to another person. You get into things like, "so you want to make a ratio where you’re comparing this corresponding side of the small and large triangle to this corresponding side" and it sounds like garbage. I felt so confusing.

She got it toward the end, but by that time we had run out of examples in the book, and when I tried to make some up, she made it clear that she felt that she understood and didn’t want to spend anymore time on it. With her, I try to make sure she truly understands, but she mostly just wants to get the assignment finished. Then I helped Nadia, her sister, with her assignment about lines of symmetry. She got it very quickly when we talked about what certain designs would look like if she held a mirror on the line of symmetry. She did about 5 examples that proved that she did understand, and she said her teacher had made it seem much more confusing. (I should have asked how he had explained it, but I didn’t think to-next time!) I think she either has a really confusing teacher or has a mental block to understanding what he explains, because it usually takes about five minutes for me to get her to understand a concept completely. She even felt confident enough that she could finish the assignment without me sticking around to make sure she was doing it right.

time 11:40-11:50 10 minutes

January 6, 1998

Tutoring Cheryl today was typical. The first words out of her mouth were, "I hate geometry." I almost don’t know how to respond to that anymore. I used to say, "Oh come on, you love it" and proceed to try to show her how fun it can be. But I don’t think she’s convinced. (Maybe she can tell that I’m not convinced myself!) Her geometry homework involved those new flow charts that have replaced the traditional geometric proof tables with statement and reason columns. She really had no grasp on the way the charts flowed or how you decided what to put where in the boxes. I don’t think she understands that the point is to figure out clues that will help her make a discovery that will be the final part of the chart. She had the hardest time grasping that whatever the problem told her to prove or show was what went into the last statement box. I don’t think her teachers ever really explained the point of the charts or what they mean in a general sense. Or even their power, because they actually do a better job of visually laying out a proof than the listed tables of my day. She also had an algebra review that involved points, lines, slopes, the distance formula, and finding equations for lines given a slope and a point, etc. One thing she kept getting confused about, and I’ve seen this a lot in Mr. Brodkey’s algebra classes too, is the fact that x can be anything. She said more than once that x is one. And when we were substituting a point and a slope into y = mx + b, she would forget about the x when she put in the slope, because she always thought x was one. I guess no one clearly explains that x stands for different numbers. Also, I think there’s confusion with the teacher saying that if there’s no number before the x, then it’s a one. But it still doesn’t mean the x is a one. The rest of my tutoring was pretty normal today, although Alisha faced the usual frustration of algebra story problems. I emphasized drawing pictures and diagrams, which she found very helpful. Erin needed help with more fuzzy subjects, so we ignored math tonight. She’s also doing the flow charts, but seems to understand them better than Cheryl does. At least she gets it that a specific reason goes with a specific statement about the figure, which Cheryl had a hard time grasping.

time 8:55-9:05 10 minutes

January 8, 1998

I help Alisha a lot tonight with story problems for algebra. When I got to her house, she said she was glad I came tonight because she didn’t understand anything in class. So we sat down and worked really hard at drawing diagrams and labeling them correctly, and then making workable equations from them. She really got the hang of it, mostly by going through some examples with me step by step. Generally I would show her one example, with some help from her but not trying to confuse her or demand too much initial understanding from her. Then I would let her have a turn, and I wouldn’t say anything unless she did something wrong. Only once did I have to stop her, and it was for a multiplication mistake, not even a process mistake. She felt really good about it. I think it helped to give her a chance to try problems without me helping AT ALL. I was literally silent. Normally I guide students through problems, and I often wonder if they could have guided themselves if I hadn’t been there. But Alisha clearly could, at least with today’s work. It was really rewarding for both of us, and at the end she sighed and said, "How come you explain things so much better than my teacher!?!" Gotta love the kid.

Then I moved on to Ben’s house. It was my first time tutoring him and he’s in 6^th grade. I think his biggest problem is that he doesn’t write out steps. Being able to do problems in your head is important in many cases, but it also leads to many careless mistakes. It was clear that it affected his performance significantly. I tried to stress it to him a lot, but he was pretty resistant, because he has his style and it works well enough, plus writing things out takes much longer. He was working on adding and subtracting fractions. He knows all his math facts forwards and backwards, unlike many kids that I tutor who still count things out on their hands or rely heavily on calculators. I was impressed and made some comment to his parents. They attribute his familiarity with math facts to Kumon (Kuman???). I don’t know tons about that way of learning math, and I would like to find out more. The other kids I’ve tutored who have done it have hated it and therefore resisted learning through that procedure. So I don’t know what’s different about Ben, but whatever Kumon is doing with him, it’s working.

time: 9:40-9:50 10 minutes

January 13, 1998

Tutoring Cheryl actually felt more promising today than usual. She was doing a review because they have finals next week, so there was no new material. That was also good for me because I was more competent and well-rehearsed at explaining it. We went through tons of problems, including proving triangle congruency, finding angle measures for different polygons (180 x (n-2)), and coordinate geometry. A lot of it seemed to sink in, since she was getting the chance to go back to material that she had already encountered. I think it raised her confidence level too, that she could do a semester’s worth of math. I tried to help her learn formulas by explaining that they made sense (and why), because there will be too many formulas for her to memorize if she just plain memorizes them. So for the slope formula, we talked about how y2 minus y1 actually means the rise, etc. For the area of a trapezoid, we talked about "averaging the bases" to find the midsegment, rather than just memorizing (B + b)/2. I think she really started to get some things that had just been crammed into her head before.

Alisha started out with her usual "I really don’t get what we have to do for tomorrow" line. She had a test to study for on story problems in algebra. Some were poster/mat, patio/garden, border problems: TOTAL AREA = INSIDE AREA + BORDER AREA. She figured those out pretty well, with the usual confusion that if there’s a 3-inch border around a poster, it adds 6 inches to the length or width, not just three. But she really mastered those problems. The harder ones to figure out were RATE x TIME = DISTANCE problems. She does a great job of filling in all the parts of the little chart she’s required to make. But she and I both got confused at the point of writing an equation. For example, we had to figure out rules, such as when two people are moving toward each other from separate points, when they cross paths they have traveled some number which is the total distance between the two points. And when two people leave the same point but at different times, the second person catches up to the first at a point where the two distances were equal. Not hard stuff, but it was challenging trying to help her learn and understand when to set up which kind of equation to solve. With Alisha, a big issue is how she feels going into a test. She gets really nervous and very conscious of how long the test takes everyone else compared to how long it takes her. So I usually try to get her to a point where she’s confident enough not to worry, and secure enough not to be as aware of the people around her. It was harder with the material we were covering today, but I think she felt much better when I left than when I arrived.

With Erin, I mostly worked on coordinate geometry. She was also reviewing for her final. She understood everything, and just needed to see it one more time to remind herself how to approach different types of problems. With her, the problem is always keeping her attention. Since she breezed through the review sheet, we mostly worked on other subjects.

time: 11:10-11:30 20 minutes

January 15, 1998

Alisha got her story problem test back today. She hadn’t done very well (a C-). She said she had felt rushed and consequently did poorly. Her teacher seems to be insensitive about time needs. That’s the kind of test that some kids just need lots of time on, but the teacher had to leave for a meeting and made them all turn the test in. She was actually happy with how well she understood the story problems, even though her grade didn’t reflect it. She got a few problems completely right, which to her proved that she understood how to do them in general. And she felt that her teacher is so stuck in her ways about grading, etc. that she didn’t feel too bad about the actual grade. She had points taken off for the stupidest things, including not writing her generic equation for each individual problem, but only once at the top. That kind of thing is frustrating because kids usually feel like they are only worth how much their score or grade says they are worth. And she works so hard, but in this case her score didn’t tell the outside world that she understood. She was a good sport though. And the new topic is properties of exponents, which is much more just number crunching. It’s harder to make big mistakes or feel time pressure. So she actually enjoyed doing the semi-monotonous homework, if only because it was more straightforward than the story problems. I worked with Ben today too. His main problem is usually not writing out enough steps and making mistakes doing computations in his head. For example, today he was multiplying fractions, but he was canceling the numbers in his head and was making lots of mistakes with his final answers. I’m still searching for a way to get kids to write out more work, especially when I’m not there looking over their shoulders.

time: 8:45-8:53 8 minutes

January 19, 1998

Today I studied with Cheryl for her geometry final. It actually went surprisingly well, especially considering that we were going through a whole semester of material she hadn’t totally seemed to grasp the first time around. It makes me wonder if things take time and distance to sink in. She actually seemed pretty confident (a rarity for Cheryl) about taking the test. She had an organized study scheme and everything. I think a lot of the time people don’t feel like they need to study for math tests, and they are so wrong! Cheryl seemed to be really motivated to study, and acknowledged how much time she would need to spend on the material in order to fully grasp it. We’ll see how it goes. I worked with Hana on division again. She definitely has some major gaps in basic multiplication times tables. It took her way too long to figure out how many times 8 would go into 27, for example, because it did not occur to her right away that 24 is a multiple of 8. Major gaps! She’s only in 6^th grade, but it seems like multiplication tables should be old news. I think another problem with her (at least with division) is that she gets confused by all the clutter of each problem instead of doing problems piece by piece. A long division problem is really a set of division, multiplication, addition, and subtraction problems (with a little bringing down thrown in for fun), but she can’t see it as little problems. To her, there are tons of numbers floating around and she doesn’t know where to concentrate her attention. The one thing she really has going for her is a great attitude. I guess she’s still too young to have math and math classroom atmospheres getting her down. Let’s hope she can remain strong and positive over the next few years.

time: 12:00-12:10 10 minutes

January 22, 1998

I helped Alisha today with more rules of exponents. It was pretty basic and she seemed to catch on quickly, and be proud of herself. One good sign is that she didn’t think she’d need me again on Thursday. Usually she likes me to come in part because it gives her confidence, and because I affirm that she’s doing everything right. But she seems confident enough with the new material that she didn’t need me again! I hope that confidence carries on into other harder units! Ben got a test back today and had gotten a C. It’s really frustrating with him, because he really understands everything. He just makes tons of careless mistakes. I’m not quite sure what I can do about it, especially since I can’t go sit next to him while he takes the test. I can only encourage him to check his work, which most students forget to do or don’t have time for anyway. I realized that I haven’t helped many boys only with math. So it should be an interesting comparison to see how he reacts to obstacles, grades, difficult concepts, etc. compared to my female math students. So far he seems not to be too affected at all by what goes on in math. We’ll see how that progresses.

time: 11:15-11:21 6 minutes

January 26, 1998

Hana is still doing division. It seems like they’ve been doing it since the first week of school since break. She didn’t have any homework, so I made up lots of division word problems for her to do. I also had her make up some word problems of her own, which she really enjoyed. And it showed that she truly understood the concept of a division word problem—which kinds of real-life situations might involve division. That’s all we did the whole time, and surprisingly she seemed to really enjoy it.

Lara and I worked on some new geometry stuff, mostly regarding geometric means. She was pretty disappointed because most of her finals brought her grades down, including math. We talked about how the first round of high school finals is difficult because students have no idea what they’ll be like, and now that she knows better what to think about (and keep—in an organized way!) she should do better next time. But she seemed pretty disappointed with herself.

time: 10:40-10:45 5 minutes

January 27, 1998

I worked with both Erin and Cheryl today, and did basically the same geometry work with each of them. Their teachers do very different jobs of explaining things and giving organized assignments. Erin’s teacher gives them the final answer to every problem on a neat, typed-out answer sheet. It really helped us confirm when we had done the problem right or not. Cheryl’s teacher hadn’t explained things very well and gave no answers to the problems in the book, so we were basically just hoping we were doing it right. I was proud of myself for understanding all the material, and all the different manipulations of information. Even I get ecstatic when certain math ideas come easy to me!

Both Cheryl and Erin had done very well on their finals. Cheryl brought her grade in the class up from a C to a B, and Erin maintained her 85%. Cheryl seemed thrilled, and I think she knew all her studying had paid off. It’s actually really good because she’s likely to do it all again next quarter. She’s learned the value of saving notes and papers and of actually studying math!

Alisha and I studied for a test she’ll have on Thursday. We hadn’t learned a lot of the material together, since I hadn’t come last Thursday, and I taught her some new ways to think about things. As usual, she felt that a new explanation was easier to follow than the one her teacher had given. She wanted me to come back Wednesday before her test to review everything again. I think she gets more confidence just from having me assure her that she’s on the right track. She definitely could pass the test without me coming back. I guess she just wants to go over things on the last night with someone who can tell her if she knows what she’s doing. Whatever will give her confidence!

time: 9:45-9:55 10 minutes