JP1 Finding the Value of an Angle

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This eighth grade mathematics lesson focuses on two-dimensional geometry - in particular, parallel lines and angles. It is the third lesson in a sequence of seven lessons on this topic. The lesson is 50 minutes in duration. There are 31 students in the class.

00:00:02[ Bell ]
00:00:04How about today?
00:00:08Good morning.
00:00:09Stand up.
00:00:16Sit down.
00:00:22Okay. It's supposed to be as usual so let's go on without being concerned about it.
00:00:32Anyway your face might not be taped.
00:00:34[ Laughter ]
00:00:35Actually there is just going to be a black line across your face.
00:00:37[ Laughter ]
00:00:39Don't worry.
00:00:53Okay, yesterday was about the relationship between parallel lines.
00:00:56We ended quickly yesterday by doing some problems.
00:01:00Do you remember what they were?
00:01:27The first written problem on the handout we hurried through it yesterday,
00:01:31so we were not able to summarize it well at the end...
00:01:35So today in between the parallel lines there is a line that is bent into an angle.
00:01:43So first we will obtain that angle.
00:01:46When I had you do the problems yesterday
00:01:48we had three problem solving methods that were presented...
00:01:51It's okay to use the method you think is the easiest.
00:01:57Please try to do the problem.
00:02:32After you have finished writing, please add an explanation.
00:02:37My- my method is this and this way.
00:02:39If you can also give an explanation that would be amazing.
00:03:36This, right.
00:03:38You're fast.
00:03:54If you forgot how to solve it, it is okay to look at the previous handout.
00:04:06(inaudible) let me borrow this for me.
00:04:09Let me borrow this because he needs a pencil.
00:04:13It's a little different... line... (inaudible).
00:04:22Umm. Okay.
00:04:26Are you done? Can I ask? How is it?
00:04:32People who solved it?
00:04:35Okay then... Arai.
00:04:42Okay, what did you do?
00:04:43Fifty. The 50 degree.
00:04:46The line around it. The line there extend it.
00:04:50Okay. Did you extend it like this?
00:04:53Bring that over there.
00:04:56Uh- 50 degrees. Alternate interior angles of it.
00:05:00There. That is 50 degrees so.
00:05:04This? Okay.
00:05:06The right angle is 30 degrees.
00:05:09If you add it all up it becomes 180 degrees so.
00:05:12A triangle. Yes.
00:05:14Yes. The top part is 100 degrees.
00:05:18The one before. A right angle is.
00:05:18A right angle is.
00:05:21Um, it becomes 180 degrees so.
00:05:25Okay. Not a straight line?
00:05:26Yes. A straight line becomes 180 degrees
00:05:29So X is 80.
00:05:32Okay. Is this okay? Here we go. Okay.
00:05:35We can see he was a little tense.
00:05:38Um, that was a clear method. People who did it using this method?
00:05:45Oh... okay.
00:05:48About five people? Okay, other methods.
00:05:52Okay, Bunya.
00:05:56Okay, please go ahead.
00:05:58In the middle.
00:06:00Draw a... parallel line.
00:06:07Oops? Okay.
00:06:08Then. Um, the top 50 degrees, the alternate interior angles of that.
00:06:15It is made there so.
00:06:17Yes, well- Well, okay.
00:06:21So it's 50 degrees.
00:06:23The bottom is 30 degrees.
00:06:25It can also be done in the same way.
00:06:28Add those together it becomes 80 degrees.
00:06:31Okay. That's good.
00:06:32The method was to use the alternate interior angles of the parallel lines.
00:06:38People who did it with this same method?
00:06:42Right. About 10 people. Okay.
00:06:47Are there any other methods?
00:06:50Oh. One person... Chika.
00:06:56Umm, from point B.
00:06:58From point B. Okay.
00:06:59From point B.
00:07:01Is it... L?
00:07:04From the top line it would be an L, huh.
00:07:05To the top straight line draw a perpendicular line. Then-
00:07:09Draw a perpendicular line from point B.
00:07:11Then. The B area is 60 degrees. Umm- well not really B,
00:07:16but between that perpendicular line and 30 degrees is a right angle.
00:07:22So subtract 30 degrees from 90 degrees.
00:07:25It becomes 60 degrees.
00:07:27The area A is fi- fifty degrees so.
00:07:32Umm, the straight line is 180 degrees.
00:07:37Angle A becomes 130 degrees.
00:07:42Since if you add up all of the angles of a quadrilateral it becomes 360 degrees.
00:07:47Umm, if you add-
00:07:51what is this? It becomes 280 degrees. Eight.
00:07:55Subtract 280 degrees from 360 degrees.
00:07:57X became 80 degrees.
00:08:00Okay. Yes. Here we go. Okay.
00:08:04People with the same method as this?
00:08:08Oh. Just Doi. Okay.
00:08:11Good. Are there any more?
00:08:14Think a little more.
00:08:17I'm thinking.
00:08:18Okay? I am sure there are people who also extended this one.
00:08:23Here are methods like that.
00:08:25Basically these three are the ones you have thought of.
00:08:29Now I will summarize these into main points.
00:08:35With just this diagram...
00:08:38with just this diagram we can't solve the problem.
00:08:41So Sano and others extended the line to make a triangle.
00:08:45Bunya and others drew in a parallel line in the middle.
00:08:49Chika and others drew in a perpendicular line.
00:08:51That means we can't solve it without auxiliary lines.
00:09:12The auxiliary lines become important.
00:09:15Then by the way we draw the auxiliary lines in all
00:09:20we got three types. Sano's, Bunya's, Chika's.
00:09:25They were different based on the way the auxiliary lines were drawn.
00:09:28That means, one way is to draw a parallel line and
00:09:31use the alternate interior angles.
00:09:37Another is to extend the line and make... a triangle,
00:09:42and use the fact that the sum of the angles of a triangle is 180 degrees.
00:09:49And Chika used the method that if we add the four angles of...
00:09:56a quadrilateral we get 360 degrees. These are the three we got.
00:10:03Now then, any method was okay,
00:10:06but the task was to do it using the easiest method.
00:10:10And each person seemed to have selected one out of the three we summarized yesterday.
00:10:14Today's problems are ones that can't be solved without auxiliary lines.
00:10:17We want to do problems of this kind.
00:10:21Today. The way this problem is made is that
00:10:25it is a problem of an angle bent once between parallel lines.
00:10:32The angle is bent once between parallel lines.
00:10:36Today by changing this condition, by changing this condition-
00:10:41I will have you make your own problems.
00:10:45And today we won't change the parallel lines.
00:10:50We won't change the parallel lines, but rather the middle.
00:10:52Change the middle means-
00:10:55Oh. It's okay with the outside. The outside can be good.
00:10:59Now it has one bend.
00:11:02It is even okay to bend it twice.
00:11:03If I say this I know there is going to be someone who is going to bend it like 10 times.
00:11:05[ Laughter ]
00:11:08Eda (inaudible).
00:11:09Eda difficult.
00:11:10There are always at least two or three of you thinking about it?
00:11:12[ Laughter ]
00:11:15I know there will be some.
00:11:18But twice or three times. That's the limit.
00:11:21After that nobody will understand it.
00:11:23You may be creative in between and change the number of bends, or,
00:11:26it is also okay to use the outside with one bend,
00:11:29or it is also okay to be outside with two bends.
00:11:32Please think by yourselves.
00:11:35You can't just go and bend it 10 times. Don't draw random squiggly lines. That is not good.
00:11:38Okay, umm, I have prepared two on your handout, but one is okay.
00:11:45If it's easy please do two.
00:11:48It is okay to bend twice or be outside.
00:11:51Then put in angles by yourselves and
00:11:55write in which angle is X, the one we need to obtain.
00:12:17Oh, if 30 students do it,
00:12:22even if they made only one problem each we will have 30 problems.
00:12:27It's actually pretty hard to put in the degrees.
00:12:30You can't just put in any degrees.
00:12:33Do we also write the (inaudible).
00:12:43When you turn it in you must be able to solve it.
00:12:47If somebody says he or she doesn't understand it
00:12:49you must be able to say what and why the problem is.
00:12:51Please think about how the problem can be understood.
00:13:10A moment.
00:13:14Wait a minute.
00:13:29Um? This one, too?
00:13:39Which one?
00:13:56Um- make one and try it yourself. Make sure you can solve it, okay?.
00:14:28These are the same, since there is only one bend between the parallel lines,
00:14:31it's the same as this.
00:14:35Try to change it.
00:14:40Oh, it looks difficult.
00:14:44It looks pretty tough.
00:14:52This also looks difficult.
00:15:03You can put the bends in, but does the number fit in, is the problem.
00:15:16Isn't it-
00:15:21It has to be a problem you can solve.
00:15:33It can be the second one or the outside.
00:15:38Bam, bam, and it can be bent on the outside.
00:15:42Or you can bend it once, twice.
00:15:47(inaudible) Just one?
00:16:01Whichever equals.
00:16:02If you make it too hard, well it'll be... stressful.
00:16:12How many times is this bent? One, two, three, four times it has been bent.
00:16:15It's been bent too many times.
00:16:22Is this okay for the first one?
00:16:24Is this okay for the first one?
00:16:26You can't solve this.
00:16:28Without another angle I don't think it can't be solved.
00:16:34If you try to solve it and you can solve it yourself then it is okay.
00:16:46Isn't it difficult? This is difficult.
00:16:55If you have finished.
00:16:57Where do you solve for this?
00:17:00Where do you solve for this problem?
00:17:01For this, it is for this.
00:17:03Then what is this?
00:17:07There's a reason for this.
00:17:09It starts at point A and goes through point B and-
00:17:12Ah, I see.
00:17:16Ahh, it looks hard.
00:17:18This is simple.
00:17:20Ahh, it is hard.
00:17:21(inaudible) like these.
00:17:25Like this.
00:17:28This looks hard.
00:17:29This is 90 degrees.
00:17:30Oh, I made a mistake. This is going to be hard.
00:17:34This 65 degrees is last.
00:17:39Eventually, if you extend it here is 62. This is 180 degrees.
00:17:43Two hundred, 220.
00:17:52Is this 62 degrees?
00:17:56Here, lend it to me.
00:18:03One hundred twenty.
00:18:10At 15 you can do it.
00:18:33Are you done writing? Are you finished?
00:19:09It seems hard.
00:19:12Let's try hard to do one problem. Try hard to make one problem.
00:19:56Um, if you put it on the inside maybe it'll be easier to solve.
00:20:01Wow. This is how you take off from A and bend and reach B.
00:20:08What does that mean for A?
00:20:13This is fine.
00:20:18Is it solvable?
00:20:19You can solve this.
00:20:20It is solvable.
00:20:22It won't be solvable without these angles here.
00:20:24What? Add this, like this?
00:20:26Like this. If you extend it... it's 60 degrees.
00:20:38This, isn't this 60 degrees, this... like this isn't 60 degrees?
00:20:42That doesn't work if this and this are not parallel.
00:20:47If you draw a parallel line.
00:20:49If you draw a parallel line.
00:20:50If this and this are not parallel then this angle is not 60 degrees.
00:20:55This angle, too... can't be used under that assumption?
00:20:59No. If they're parallel lines then just put in the symbol for parallel lines here. (inaudible)
00:21:16Thank you very much.
00:21:19I don't know if they'll understand this.
00:21:21It looks a little tough. This is too difficult.
00:21:25And right... I made it a little too difficult. I made it too difficult.
00:21:32Here it's hard.
00:22:06If there isn't an angle here it can't be understood. Just put something in.
00:22:21You guys seemed to have learned that any number just doesn't fit in.
00:22:25You guys seemed to have learned that any number just doesn't fit in.
00:22:46How many bends are there? One, two, three, four, five bends.
00:22:50If you don't make it at most three then it will be impossible.
00:22:53I understand.
00:23:11Let's see. What time?
00:23:12Well it seems it was a little hard. I made a mistake.
00:23:16There are many of you that are in trouble.
00:23:20Well there are not that many people with one, two done,
00:23:22but there seems to be a lot of people with one done,so,
00:23:24get in your groups and... from the problems you have made...
00:23:33pick a problem you and others think is challenging,
00:23:38and group leaders please bring them up here.
00:24:01Please check if the problem can be really solved and then bring it up.
00:24:21One problem is enough.
00:24:23There are problems that can't be solved.
00:24:43How about this one? This?
00:24:46Th- Th- This (inaudible).
00:24:57(inaudible) It seems hard.
00:25:03Can't it be solved?
00:25:06You don't have to make a new problem at this point.
00:25:09Just from the currents ones you have.
00:25:19This is a good problem.
00:25:20Is it good?
00:25:23Yes, because it's two bends, and the bends are different.
00:25:27And this is also a good problem, whether you can do it or not,
00:25:30change that part. Not too much.
00:25:36Up to number two?
00:25:56(inaudible) a quadrilateral.
00:25:59Ha ha. Indeed.
00:26:01A quadrilateral. A quadrilateral? (inaudible)
00:26:13Who made this?
00:26:20You can't solve it.
00:26:23Mmm. Can they do it or not?
00:26:54Hey, let's bring up one.
00:27:04Ha, it looks like a hard one.
00:27:05Whose is it?
00:27:07From which group?
00:27:08From group number five.
00:27:09This one?
00:27:11Group five.
00:27:36It's a hard one.
00:27:42Oh, the diagram was too small.
00:27:45You're completely wrong. Do you know what you are doing?
00:27:50This part is easy.
00:27:56Here teacher.
00:27:57This- which group?
00:27:58This one.
00:27:59This can't be used.
00:28:00It can't?
00:28:01Uh huh. Because this is bent many times.
00:28:03Teacher. This one.
00:28:04Just a little more- this is too hard.
00:28:06He said it was too hard.
00:28:08Oh, it looks like a good problem. What group?
00:28:11Group one.
00:28:12Group one?
00:28:19Group one?
00:28:20Group one.
00:28:27The diagram is a little...
00:28:56Which number?
00:28:57Number four.
00:28:58Number four- can you really solve it?
00:29:01That can be solved easily.
00:29:02We can.
00:29:03It seems.
00:29:06Is this all?
00:29:34Okay, that's good. Yes.
00:29:38Next, who's next?
00:29:41It's easy. That one.
00:29:44Don't say anything yet. Don't say anything.
00:29:47Groups that have not yet brought up one, please hurry up.
00:29:54Teacher, I solved this.
00:29:57I wonder.
00:29:58It can- it can be solved teacher.
00:30:01Can this be solved?
00:30:02I believe it can be solved.
00:30:06I think it is impossible.
00:30:09This one, this one.
00:30:10Which one?
00:30:11This one, this one. (inaudible)
00:30:13Mm. It's probably right?
00:30:19Which one is that, group three?
00:30:20Group three.
00:30:26The second time is hard but easy.
00:30:37Number one is easy, too.
00:30:50Now number four is left.
00:31:01Which number?
00:31:02Number four.
00:31:03Number four?
00:31:04Number four. It might be complex. Right, it's different.
00:31:06It might be complex, okay?
00:31:08I'll do number four just a little. Sorry, okay?
00:31:32Can it be solved?
00:31:33It can be solved.
00:31:34You're able to solve it? Hum
00:31:36Hum oh.
00:31:42What number?
00:31:45Hum. Okay.
00:31:48They are problems that we don't know if we can solve or not until we try.
00:31:54I'm sorry for taking so much time,
00:31:57but there are six problems on the right side.
00:32:06(inaudible) if we can solve it or not.
00:32:11I wonder.
00:32:20It seems impossible to do all of them in this time so.
00:32:23So we'll think about them a little next time, too.
00:32:36Please hurry up and copy the six problems.
00:32:44Imai is X here?
00:32:50Hum. (inaudible)
00:32:54Oh, I got it.
00:32:57Can you solve it?
00:32:58I got it. I got it.
00:33:09They really seem challenging, huh?
00:33:12There seems to be one that is too challenging.
00:33:19They are really challenging, like eating a dried squid.
00:33:24[ Laughter ]
00:33:27Umm, this one is not that bad.
00:33:37Umm... I'll say it's like a piece of gum.
00:33:47Oh, group two's, group two's, the one on the top.
00:33:50Umm. Group four.
00:34:01This is a rock.
00:34:03[ Laughter ]
00:34:05Then this is a rock.
00:34:31Umm. Um um um.
00:34:52Rice cracker.
00:34:53Number five. Number five.
00:34:55Isn't it easy?
00:34:59Rock, it's a rock.
00:35:05Then I'm sorry there isn't much time, but there are six problems.
00:35:10There may be ones from your own group. There may be ones that are yours.
00:35:12Please copy them.
00:35:15It can be from number one. It can also be from the hardest one.
00:35:21(inaudible) This is gum.
00:35:46That means since today's problems can't be solved without auxiliary lines...
00:35:50what kind of auxiliary lines. It can't be solved without auxiliary lines.
00:35:57If you need the auxiliary lines,
00:35:58the problem is what kind of auxiliary lines should you use.
00:36:03There were three, but which one is the easiest way to do it.
00:36:29For now please finish copying the problems by 35 after the hour.
00:36:35I think... the answers will be given next period.
00:36:41What shall I do?
00:36:44Can you solve this?
00:36:45I don't know.
00:36:49This is well... this is really hard.
00:37:10Genji's symbol is good. It's there nicely working.
00:38:40Are there any groups that were able to solve them?
00:38:43Plus 15.
00:38:48(inaudible) since it's 15 degrees, so 90. Ooh.
00:38:52Not so (inaudible).
00:38:54Oh- the rice cracker hard problem seems to be solved.
00:38:57(inaudible) fifteen. It's 30 degrees.
00:38:59Teacher. Fifteen.
00:39:01Fifteen, 15, 30, 90, 90.
00:39:02It's not 90, 90. Ooh.
00:39:04One hundred.
00:39:11It's correct. We got this. We solved this.
00:39:19The rock?
00:39:20Later. We'll check it later.
00:39:23We'll check it later.
00:39:31No, at the time- at the time I was solving it I was thinking that I couldn't solve it.
00:40:14I didn't write the group names.
00:40:35Since all of them are challenging we can't solve it just like that.
00:40:39Let's do it step by step.
00:40:43People who are done solving the problem from group one?
00:40:47Okay good. Group two's?
00:40:50A little more. Straight. Raise your hands straight. Ahh, about the same.
00:40:54Group three's?
00:40:57Oh. Group four's?
00:41:00Okay. Group five's?
00:41:03Okay. Group six's?
00:41:05Really. Group six's. Okay good.
00:41:08Number six is easy so-
00:41:12There is an even harder one.
00:41:14This one seems more likely to work out.
00:41:22I haven't gotten this one yet, but-
00:41:26It's hard?
00:41:27I just finally got it.
00:42:04Is it group one or group two?
00:42:05One hundred eighty degrees?
00:42:07Is this 40?
00:42:09Forty degrees.
00:42:15One hundred twenty degrees.
00:42:16One hundred twenty degrees.
00:42:18Why is that? (inaudible) Isn't this 47? Right here.
00:42:36This is easier. Umm. This one symbol. If you miss that one symbol...
00:42:43you can't solve the problem. This is parallel.
00:42:46This and this are parallel. Without that you can't solve it.
00:42:54That's right. That's right. That's right. If you do that. It's about done.
00:43:23Group one's is hard. I think group one's is really hard.
00:43:48How you draw the auxiliary lines makes the difference. Something...
00:43:57One hundred ninety. This is 190.
00:44:07Number eight. Number two. Number eight. Number eight. (inaudible)
00:44:19Okay, do you want to try it again?
00:44:28I'll give you a hint. Try extending this further.
00:44:32Try extending the auxiliary line further.
00:44:53Try this one. This one is easier.
00:44:59There is a symbol included here. This parallel symbol.
00:45:05Extend the auxiliary line all the way to here. Try extending it.
00:45:15Yes, yes.
00:45:18This and this are parallel. If you use (inaudible) angles this angle becomes the (inaudible).
00:45:25This is a straight line so you know it. So you somehow get this angle.
00:45:56You know this angle and this angle (inaudible).
00:46:00If you write in as much as you'll work.
00:46:18Oh that's good. They're the right auxiliary lines.
00:46:24Umm. Ahh, this is (inaudible). This is hard.
00:46:27You can't solve it without auxiliary lines.
00:46:30That's good.
00:46:33Umm, 120 degrees and then the straight line.
00:46:40Like this line and like this, use auxiliary lines and try drawing it.
00:46:44It's good to draw auxiliary lines.
00:46:48Man. This one is amazing.
00:46:58Umm, I see.
00:47:24Do it one more time.
00:47:25I did it once.
00:47:26A mistake.
00:47:27Uh I thought it was wrong. (inaudible)
00:47:29Umm um. That it. That's it. That's it.
00:48:07(inaudible)? (inaudible)
00:48:12Umm, that one.
00:48:18Umm, um, um. Divide it around here. It's the auxiliary lines.
00:48:23Auxilary lines. It's a lot different with another auxiliary line.
00:48:29But even if you didn't get the last one, you got this one, right?
00:48:33This and this are parallel,
00:48:37and here and here are corresponding angles, and you get here.
00:48:40You got there and with that...
00:49:10This is going to be a little bothersome but, well, I want to know the present situation.
00:49:14People who are done with group one's?
00:49:18You solved it?
00:49:22Umm. Group two's?
00:49:27Oh, okay. Group three's?
00:49:32Um, okay. Group four's?
00:49:35Up to where did you get to? Um. Group five's?
00:49:40Hum. Two people. This was a hard one.
00:49:42Umm. So that's it. Group six's?
00:49:45Group six's was simple.
00:49:46It was very troublesome.
00:49:49Did you do it?
00:49:50(inaudible) Okay then there are a lot of people who are using triangles.
00:50:00[ Bell ]
00:50:01That's okay, but- There were three types of auxiliary lines.
00:50:06There are other easier methods of solving... using other types of auxiliary lines.
00:50:11We will check these in the next period.
00:50:16Return your desks.
00:50:33You can never do this one.
00:50:38Who got group four's? Who said he or she got group four's?
00:50:41Fujita Today. Okay.
00:50:46Please end the class.
00:50:48There's no math class tomorrow so let's check them the day after tomorrow. Okay.
00:50:50Do we have math the day after tomorrow?
00:50:53We have math class the day after tomorrow right?
00:50:58The day after tomorrow is Thursday.
00:51:02Did it change?
00:51:04The day after tomorrow is Friday.
00:51:07Today is what day of the week?
00:51:10Today is Tuesday.
00:51:13Day after tomorrow?
00:51:15That's right. I got it. Okay.
00:51:17Go ahead.
00:51:19Stand up. Bow.
00:00:00日本の数学教育に関する一般的なコメント 日本の中学校では数学は必修教科であり,全員が同じ数学の内容を学習することになっている。内容は文部科学省の発行する学習指導要領によって規定されている。諸外国のいくつかの国にみられるような,能力や進路の違いによる複数の数学コースは,国が定めたカリキュラムには設けられていない。学習指導要領は約10年ごとに改訂されている。日本の授業がビデオ撮りされたのは1994年-1995年であるが,2002年から新学習指導要領に基づいた授業が行われている。新学習指導要領は目標に「数学的な活動の楽しさ」という文言が入り,「自ら課題を見つけ,自ら学び,自ら問題を解決していく」数学学習の実現をなお一層目指す一方で,時間数の削減(週4時間から週3時間へ)や内容の厳選が行われた。当時中学校2年の内容であったもので,現在では削除されたもの(「近似値,二進法,流れ図)),より上学年に移行したもの(「図形の相似)は中学校3年へ,「一元一次不等式)「三角形の重心)「資料の整理)は高校数学Iへ),より上学年から移行されたもの(「確率の一部)は中学校3年から2年へ)がある。中学校数学の教科書は6社から発行されているが,いずれも文部科学省の検定を受けたものであり,題材の取り上げ方等に若干の違いはあるが,単元の配列など大枠は似かよっている。学校は学習指導要領や教科書を元に学校の教育課程を計画する。従って,日本では同一の時期には全国でほぼ同じ内容が指導されているといえる。生徒の94%は公立中学校にかよっている。残り6%の生徒は入試を受け国私立中学校に入学する。指導は一斉指導が基本であるが,複数の教師によるTT(テイーム・テイーチング)や班学習も適宜取り入れられている。2002年からは,生徒の習熟度別授業も比較的多くなっている。多くの教師は机間指導を大切にしており,個別指導を行うとともに,全体に聞こえるようにヒントを与えて思考を促し意欲を喚起しようとすることが多い。さらに,机間指導をしながら,指名する生徒や考え方を取り上げる順番を考えている教師も多く,それが,よい一斉授業につながっている。このように,教師は生徒の意見や考えを生かしながら授業を行っている場合が多い。しかし中には教師主導の授業もみられる。特に学年が上がるに従いこの傾向は強くなる。なお,TIMSS1999のデータによれば,数学あるいは数学教育を専攻し教員免許状を持った教師の割合は日本は非常に高く,このような教師のもとで学習している生徒の割合は93%(国際平均値は73%, TIMSS 1999 International Mathematics Reports, Boston College)である。
00:08:22多様な考え方, 補助線: この授業では3通りの考え方を取り上げているが,特に図形の授業では,多様な考え方を積極的に取り上げてその考え方を共有していくことが多い。また,この授業のように,図形では補助線を引いて考えることを大切にしている。
00:13:55机間指導,自力解決: この授業で,生徒はかなりの時間,自分なりに問題づくりに取り組んでいる。日本の数学の授業では,自分なりに考える自力解決の時間を大切にしようとする傾向が強まっているが,この授業ほど多くの時間を自力解決に使うことは少ない。問題づくりの授業においても同様である。なお一般には,机間指導をしながら,一斉指導の場面に戻ったときにどの生徒を指名するかやその考え方を取り上げる順番を考えている教師が多いが,この授業では生徒の考えを取り上げることはせず,自力解決後,班学習に進んでいる。
00:49:10問題づくりの授業のパターン: この授業では,何班の問題が解けたかを挙手によって確認して本時の終わりとしており,問題の比較検討や解決の段階はない。一般に問題づくりの授業では,その時間中に生徒が作ったいくつかの問題を学級全体で解決すること,さらにはつくった問題の類似点・相違点など話し合わせることが多い。
00:00:00General Comments on Japanese Math Education In Japanese junior high schools, mathematics is a mandatory subject and all students are supposed to study the same content. This content is regulated by the Guidelines for the Course of Study issued by the Ministry of Education, Culture, Sports, Science and Technology. In the curriculum designed by the Ministry, there are no additional math courses where students might be grouped in accordance with their abilities, nor on their orientation towards their future academic majors, such as can be found in other countries. Guidelines for the Course of Study is revised every 10 years. The Japanese lessons that we analyzed were videotaped from 1994 to 1995. However, current math lessons in Japan are based on the New Guidelines for Course of Study,which were issued in 2002. In the New Guidelines for Course of Study, the Ministry has inserted the phrase "enjoyment of mathematical activities" as their goal, and while they tried to achieve math learning in which students "find the task by themselves, learn by themselves, and solve the problems by themselves," they also reduced the instruction time from four hours per week to three hours per week and restricted the teaching content. At the time of videotaping, the contents which were taught in the eighth grade, but deleted in current curriculum include: "approximate value", "binary scale", and "flow chart". The content which is now taught in the higher grades includes: "similar figures", which was moved to the ninth grade; and "linear inequality with one unknown", "centroid of a triangle", and "organizing data" all moved to high school Math I. The content which was moved down from ninth grade to eighth grade includes a discussion of "probability". Textbooks for junior high school math are published by six publishing companies, and all are examined by the Ministry of Education, Culture, Sports, Science and Technology. There are minor differences among the textbooks regarding how to present a given topic. However, the arrangement of chapters, et cetera, are all more or less similar. All schools plan their educational curriculum based on the Guidelines for the Course of Study and the textbooks. Therefore, it is possible to say that in Japan, more or less the same content is taught throughout the entire school system at any given time of the year. Ninety-four percent of Japanese students attend public school. The other six percent attend a private or a National school. The basic teaching style is whole-class instruction, but lately some schools have started incorporating "TT" (team teaching) by several teachers as well as group study. Starting in 2002, there's an increasing number of ability-grouped classes (class based on students' degree of mastery of the subject). Many teachers value "kikan-shido"- i.e., strolling among students' desks while checking the students' rates of progress during deskwork - and while they personally assist individual students with the problems they are working on, such teachers often give a hint to the whole class in order to help the students' developing their thinking and increase their motivation. Moreover, many teachers stroll among students thinking about who should present the ideas, or in which order the ideas should be taken up, and this leads to a good whole-class interaction. Thus, many teachers make the most of the students' opinions and individual thinking. However, we also observed many teacher-fronted classes. This tendency becomes stronger as grade levels become higher. Also, according to the 1999 TIMSS data, in Japan, the ratio of math teachers who received their teaching certificates in mathematics or mathematics education is very high, and 93% of the students are instructed by such teachers (as compared with the international average, which is 73% (TIMSS 1999 International Mathematics Reports, Boston College).
00:04:32About the problem This problem of "parallel lines and angles" is dealt with in all the textbooks (published by six different publishers) at the time of videotaping (1994), as well as in the year 2002. Reasons given for the inclusion of the problem include: "students should be able to solve this problem utilizing the already studied material;" "it has potential to develop related skills;" "there are many different ways to think about it;" and "almost all the teachers deal with this problem in their lessons." It appears often in quizzes and entrance exams. It is one of the typical problems in geometry.
00:08:22Various ways of thinking - auxiliary line In this lesson, the teachers bring up three different ways of thinking. Especially in the geometry lessons, teachers present various ways of thinking and share their methods with the class. Also, in this lesson, they recognize the value of different ways of thinking about drawing auxiliary lines.
00:10:36Problem creating A lesson in which students create problems, change the conditions, introduce them to each other and solve them, is called "a lesson to create problems." The purpose of such lessons is to increase students' creativity. "Lessons to create problems" are conducted sometimes at the end of a theme, or at the end of the semester. They are also conducted in "target study" (kagai-gakushu) and in "mathematics as an elective subject" (sentakukyoka toshiteno suugaku) in which they were placed as of 1989's Guidelines for the Course of Study. They are not necessarily conducted on a daily basis.
00:13:55"Kikan-shido" (literally: between-desk-instruction) Strolling among students' desks, checking their rates of progress during deskwork or assessing their progress during seat work,"jiriki-kaiketsu" to solve one on your own. In this lesson, students spend a fair amount of time in creating their own problems. In Japanese mathematics lessons, there is an increasing tendency to value time for "jiriki kaiketsu" ("learn to solve on your own"). However, it is rare that so much time is spent on this, as is done in this particular lesson. Even in a typical "creating problems" lesson, not this much time is spent on "learning to solve on your own". In addition, there are generally many teachers who use this between-desk-instruction time to think about which students to select, or the order of questions and responses to deal with, while they stroll among the students' desks and this leads to a good whole-class instruction. In this lesson, the teacher does not take up the students' ideas, but simply allows them sufficient time to solve the problems themselves and then to move on to the group study.
00:23:24Group Study There are cases where the students work in groups during the math lesson, as in this lesson. However, many times the students bring their desks back to the position facing the front again and come back to the whole-class instruction. Also in Japanese classes, often times students are assigned to a "daily activity group" (such as bringing lunch or cleaning blackboards) and students may work within that group during the lesson. Other times, a special group for just the math lesson is assigned as well.
00:49:10Pattern of a "creating problems" lesson In this lesson, the teacher ends the current lesson by letting the students raise their hands in order to confirm which groups have solved the problems. In general, during a "creating problems" lesson, the whole class often will try to solve some of the problems that their classmates have created, and discuss the similarities and differences among the problems and their solutions during the lesson.
00:00:53ここで教師は学生に、昨日何について勉強したかを思い出させ、今日同じトピックで進めるということを説明している。実際、この授業は全て既習事項の復習である。このようなタイプの授業はきわめて稀である。日本のデータを見ると、授業全体が復習にのみに使われるのは全体の5%にすぎない (Hiebert et al., 2003, Teaching Mathematics in Seven Countries: Results from the TIMSS 1999 Video Study [以下 Video Report], figure 3.9) 。
00:04:35生徒は3つの方法のどれを使ってもこの問題が解けるということを明示される (1:46参照)。それぞれの方法が別の生徒によって発表される。各方法が比較され詳細に検討されているので、“方法の検討”の定義にあてはまる。生徒に解決方法の選択が明示的に与えられる問題は日本の授業の31%を占める (Video Report, table 5.2)。複数の方法がクラス全体に示される場合は授業の42%を占め(Video Report, table 5.1) 、24%の授業で他の方法が詳細に検討されている (Video Report, table 5.3)。
00:08:29ここで教師は3つの異なる解決方法を比較分析しながら問題を要約し、それぞれの場合において生徒が接線を引いたことに注目させている。日本のデータセットでは、各授業につき、平均で27%の割合で問題が要約されている (Video Report, table 5.4)。この問題は生徒が数学的に推論することが必要であり、数学的な発想を結びつける必要性を示唆するように提示されており、クラス全体で問題を解決しながらそのような推論と関連付けが話し合われる。特に生徒が示した異なる解決方法を、教師が包括的な数学の原理に結び付けている。日本の授業では問題の平均54% が“関連付け”を示唆するように提示されていると分析される (Video Report, figure 5.8)。そしてこのうち48% が実際に“関連付け”を行うことによって解決されている (Video Report, figure 5.12)。
00:10:14授業のこの時点において教師は補助線を引かずに類似問題を解くという、最終的な目標を提示する (Video Report, figure 3.12)。
00:23:11生徒はここで個人の作業から小グループでの作業に移る。この授業では時間の78%が一人一人が作業する時間にあてられている。このうちの35%は個人作業、65%がグループ作業にあてられている。これらの割合は日本のデータセット全体から見るとむしろ例外的と言える。授業の平均では、授業時間のうち34%のみが一人一人の作業にあてられている (Video Report, 図3.15)。さらにデータセット全体では、生徒はほとんど自分一人で作業をしており、授業あたり一人一人の作業は平均で76%個人の作業に使用されている (Video Report, figure 3.10) 。
00:31:48教師はここで6つの問題のセットを与える(生徒はクラスメートが作ったばかりの問題を解くことになる。この授業では全部で2つの独立した問題(単一の問題として提示されたもの)と、6つの問題群(セットとして提示されたもの)があった。独立した問題には授業時間の約60%が使われた一方、問題群には37%が使われた。日本のデータセットでは、一回の授業につき平均3つの独立問題が扱われる (Video Report, 図3.6)。平均で授業時間の64%が独立した問題に、16%が問題群に使われている (Video Report, figure 3.4)。この授業では、生徒に既に学習している概念と手順を新しい状況に応用させるものであるから、全ての問題が応用である。ここでの特定の応用問題は、実生活の設定におけるものとしては提示されていない。このことから、これらの問題は日本の授業で行われているのとほぼ同様であるといえる。データセット全体から見ると、授業につき74%の問題が応用であり (Video Report, figure 5.6)、実生活に結び付けられるのは授業の9%にすぎない(Video Report, figure 5.1)。
00:00:53Here the teacher reminds the students what they worked on yesterday and says they will continue with this same topic today. In fact, the entire lesson is a review of previously learned material. This type of lesson was quite rare. In the Japanese data set, only five percent of lessons were entirely devoted to reviewing (Hiebert et al., 2003, Teaching Mathematics in Seven Countries: Results from the TIMSS 1999 Video Study [hereafter Video Report], figure 3.9).
00:04:35The students are explicitly told that they can solve this problem using any of three methods (see 1:46). Each method is presented by a different student. The methods are compared and examined in some depth, thus meeting the definition of an "examining methods" problem. Problems in which students were explicitly given a choice of solution methods were found in 31% of Japanese lessons (Video Report, table 5.2). Multiple methods were publicly presented in 42% of the lessons (Video Report, table 5.1), and alternative methods were examined in depth in 24% of the lessons (Video Report, table 5.3).
00:08:29Here the teacher provides a problem summary, as he analyzes and compares the three different solution methods, noting that in each case the student drew an auxiliary line. On average, 27% of the problems per lesson in the Japanese data set were summarized (Video Report, table 5.4). This problem is presented in a way that implies students will need to use mathematical reasoning and make connections between mathematical ideas, and in fact such reasoning and connections are discussed as the problem is solved publicly. In particular, the teacher connects the overarching mathematical principles to the different solution methods presented by students. On average, 54% of the problems in a Japanese lesson were coded having a problem statement that implied "making connections" (Video Report, figure 5.8). Of these, 48% were solved by "making connections" (Video Report, figure 5.12).
00:10:14At this point in the lesson, the teacher introduces the main goal: to solve similar problems without drawing an auxiliary line. Seventy-five percent of the lessons in the Japanese data set contained a goal statement (Video Report, figure 3.12).
00:23:11Students now shift from working individually to working in small groups. In this lesson, 78% of the time is spent in private interaction. Of that time, 35% is devoted to working individually and 65% is devoted to working in groups. These proportions are rather different compared to the Japanese data set as a whole. On average in the Japanese lessons, only 34% of the class time was spent in private interaction (Video Report, table 3.6). Furthermore, across the Japanese data set students worked mostly by themselves; on average, 76% of the private interaction time per lesson was devoted to working individually (Video Report, figure 3.10).
00:31:48The teacher now assigns a set of six problems (students are supposed to complete the problems just created by their classmates). Altogether in this lesson there are two independent problems (that is, problems presented as single problems) and six concurrent problems (that is, problems presented as a set). Working on the independent problems takes up approximately 60% of the lesson time, while working on the concurrent problem takes up 37%. On average there were three independent problems per lesson in the Japanese data set (Video Report, table 3.3). Sixty-four percent of the lesson time, on average, was devoted to independent problems and 16% was devoted to concurrent problems (Video Report, figure 3.4). All of the problems in this lesson are applications, since they ask students to apply to a new context concepts and procedures they have already learned. These particular application problems are not presented in a real-life setting. In this regard, the problems are similar to most of those presented in the Japanese lessons. Across the data set, 74% of the problems per lesson were applications (Video Report, figure 5.6). Only nine percent of the problems per lesson contained a real-life connection (Video Report, figure 5.1).