RJ:
Today we’re going to be doing lots of investigating activities. And
do you remember when we talked about the warning signs that we walked
about a couple of days ago. Who remembers some of those warning signs?
Student: The school crossing sign was in the shape of a pentagon and it had acute angles and right angles and obtuse angles, and some of the lines were parallel.
RJ: My name is Retella Jones, this is my 6th grade math class. In our system, in Durham public schools we adopted a math curriculum two years ago that we felt would address student understanding of mathematical competencies. We had seen students in our system struggling with math, they were not being successful with rote memorization and Mathscape gave us a way to present challenging and meaningful math to the students that would engage them and help them develop their critical thinking and problem solving skills.
RJ: Today we’re going to be doing some more exploring. And we’re going to start off by having a group come to the front and I want you to use ribbon, we’re going to use ribbon this time to explore and I’m going to give you some directions to do this particular activity. And if everybody can sort of pull back, we’re going to need to stand up. This group needs to stand. You want to make a shape with four sides that has only one pair of parallel sides.
Student:
This has five sides, so what we’d have to do is we’d have to stretch
it out more.
Student: Keep this straight, but try and make sure that these two is a trapezoid.
Student: Yes.
Student: A trapezoid.
Student: If you pull this out a little more and Stephanie, pull your side out, pull it out like this, that’s more of a square. We need to make a trapezoid type of object. Stephanie, you need to make your side straight, like pull, like really hard.
Student: That works. That’s four sides and two are parallel.
RJ: So I’m going to ask the students that are sitting around, to get up --- and let’s look at the shape that Megan’s group has formed and see if it meets all of the characteristics that I stated. They are making a shape with four sides that has only one pair of parallel sides. Okay, Trip, does it have four sides?
Trip: Yes.
RJ: Would you point to the four sides and count them for me.
Trip: 1, 2, 3, 4.
RJ: Okay, can someone identify the parallel sides for me? Okay?
Student: Here and here.
RJ: So did their shape meet the characteristics that I asked them to use to form that shape? Okay? Well, that’s exactly what we’re going to do now in our small groups. You’re going to make some shapes, I’m going to give you some clues and your shapes must meet the characteristic in those clues. If the shape does not work and you cannot do it, you are going to have to say it’s impossible to do and you’ve got to tell me why.
RJ: The ribbons are concrete objects that they can touch and feel and move around and they can change the shape without having to erase and change a picture on a sheet of paper. So if they have opportunities to work together and move this shape around with the ribbon, and not have to do it by themselves, they explore many options and they question themselves and each other.
RJ: So what I want you to do, in your small group, using your ribbon, I want you to first make the shape.
SJ:
Students probably could do activities like these on the computer using
pencil and paper, but there’s so much more power for the student in actually
physically getting their hands on it and holding the ribbon in the group
and being a part of that shape and having to really work out, where do
we need to be to make this shape fit the clues that we’ve got. Middle
school students are very visual and very kinesthetic learners so the more
opportunity we could build into the curriculum to get them up and doing
very hands-on kinds of things so that they would learn it both visually,
orally and physically, we looked for those opportunities and tried to
build them into the curriculum and this is one of those places where we
saw an activity that really lent itself to having kids standing up, moving
around, as a way of really strengthening their mathematical understanding
about the concepts they were learning.
Student: We know that we need the shapes, at least two pairs of parallel sides.
Student: And we automatically know that it’s not going to look anything similar to a square.
Student: Yes, so this one would be parallel, these two will be parallel, because they are right across from each other. This one, this side and this side would be parallel because they wouldn’t run into each other.
Student: Well probably eventually that one would, because the way it’s slanted and then this one would be going straight up so it would go slanted –
Student: And this one also would be parallel.
Student: Yes, that’s what I was saying, like this one and then this one and then this one and this one and this one and that one.
Student: Because we have to have at least two and we have one more than two parallel (?).
Student: Yes, okay, now (four?) sides.
Student: So these two, and these two can be parallel too can’t they, because they are right across from each other.
Student: No, because they wouldn’t hit each other.
Student: Those are sort of slanted and if they are parallel then we have to make these two straight too and then it will be equilateral.
Student: That’s still parallel right there.
Student: Well, yeah.
Student: Okay, I’m saying it’s like (Kim?) pull your side up, with your hand over there, yeah, like that. So then we have the 1, 2, 3, 4, 5, 6. See the 6 sides?
Student: A hexagon.
We were looking for a series of investigations we could develop that would push kids beyond that simple identification of geometric shapes to thinking about the properties of the sides and the angles and how those interacted with each other.
Student: This one and this one would be parallel.
RJ: When students have had geometry work prior to 6th grade, they’ve often seen shapes, they’ve learned the names of shapes, they can probably identified sides and sometimes identify angles. And they can measure them, but rarely do they go the step beyond to really look at how those properties affect each other in certain shapes. For example, rarely will they get to the point of saying, look, if I have two sets of parallel sides, the opposite angles are going to have to be equal and it can’t be otherwise and to really relate the properties of sides and the properties of angles.
Student: We need an equilateral shape with more than three sides and no parallel sides. Now how are we going to do this you guys?
Student: Well, so let’s go ahead and start out with four.
Student: But then all of them are parallel.
Student: So let’s try five.
Student:
(?) pentagon and one of them (?)
Student: Then we could, but if we, we could move –
Student: It wouldn’t be an equilateral.
Student: Yeah, but if we can make it like that.
Student: Yeah, but then it wouldn’t be equilateral, because that is longer than (?).
Student: No, but you could do it like –
Student: It won’t(?) be equilateral –
Student: But then that part would not be equal and this part right here is not, it’s a lot longer than most of those.
Student: But you could move this one up and then it would be half.
Student: But then that won’t apply over there.
Student: Well then move it.
Student: It’s not as long as that.
Student: Well we can make it like that.
Student: Well, but still it’s not an equilateral.
Student: It’s still longer (?)
Student: It’s still not equal.
Student: Yeah, because this one’s longer than all of those, so that’s impossible. I think it’s impossible.
Student: Yeah, I sort of agree with impossible, because any way you make it you can’t make this one.
Student: It won’t be equilateral in their parallel (?)
Student: Yeah.
Student: That’s impossible.
Student: So – it’s impossible.
Student: Okay, it’s impossible.
Student: I’ll write impossible.
Student: It’s equilateral still.
Student: Because all the sides equal –
Student: If you have equilateral sides then they are going to have to be parallel in a way.
Student: An equilateral shape without parallel lines.
Student: Does it really make any sense?
Student: So it would have to be, some sides would have to be equal and it would have to parallel.
RJ: Okay, how many people had clue one? How many groups had clue one? Keep your hand up if you found a solution. If you found a solution. Two groups found a solution. So the other two groups, you think it’s impossible? You think it’s impossible. Okay, put your hands down. What you can do, this is what I want you to do, because two groups found a solution. I want you to select one person from your group to go like be a scout, and come to other, you can choose either group you want, maybe this group in the back, they had a good solution and get a hint. They are not going to tell you how they got the answer. They are going to give you a hint and you go back and share that with your group and see if you can come up with a solution.
Student: If an equilateral shape is more than three sides and no parallel sides so –
Student: And it has more five sides, the one that we have. It has five sides. Because if it has more, because we know that an equilateral triangle is three sides with no parallel lines. And so we thought maybe if we tried it with another odd number, because we know that most our even numbers will have at least one or two parallel sides and so we try it with another odd number, the next to small odd number was five, so we tried it with five sides, but we’ve also been trying it with 21 and 7 and all those big ones, and before we got it, we also thought it was impossible, but then we finally, we were just chitchatting around trying to figure out what it was and we just were messing around, we just like got it. So do you want to try and see what you think the shape is?
Student: (?) parallel lines.
Student: We’ve just got to give them a clue.
Student: Think of diagonals. Like a diagonal line. Like it’s going to be all diagonal except for one’s going to be either, I guess, depending on the way that you do it, it’s going be horizontal, it’s going to be all horizontal and (then it’s going to go up?) or anything.
Student: So it’s going to be a twisty-turvy shape, it’s not going to be anything normal looking.
Student: It’s going to look like a trapezoid sort of. But one side is going to move in like that and it’s going to have one side that looks like this.
Student: Then it won’t be an equilateral, because that side would be obtuse.
Student: Well all the sides will be equal.
Student: All the sides will be equal, I’m just drawing it like –
Student: Once you fold it up and make –
Student: I know, but once you fold it up and measure it all with the string, then it all ends up being equal.
Student: Make it seem like what it should look like. You’re going to bring one side into the center.
Student: You wouldn’t measure it if you take a point in the string, and bring it over, and so these sides match up where her point is and they all match up that way.
Student: But this has to be a tad bit longer, because this (?) take up three sides, so it’s got to be the sides of three sides, so this has to fold into thirds, I think.
Student: And this is equilateral.
Student: So it seems this folds into thirds in some way shape or point, then you can start moving it around into(?) and get your shape. You can come take my side, if you want.
Student: One of the sides goes out and one of the sides go in, but it’s not parallel.
Student: What do you think the shape looks like at the moment?
Student: Something like that?
Student: You tell us what to do.
Student: I’ll hold onto it.
Student: Hold it –
Student: Where do you want us to put our hands? Do you want –
Student: Do you want that to go in?
Student: You might want it --
Student: Like that?
Student: Yes.
Student: Now, do all the sides look equal to you?
Student: No.
Student: Do you want to measure?
Student: Okay.
Student: Okay, we’ve got to bring yours in. That’s equal. Now bring –
Student: And then that to that is equal and then –
Student: And those to those are equal. They were all equal.
Student: The side at the bottom wasn’t.
Student: They were all equal. The sides at the bottom were equal.
Student: So you had a shape similar to that.
Student: Bring that closer and then you bring that out and it looks like that.
Student: Because that’s what you got, you got us to do. That.
Student: So do you understand it?
Student: Yes.
Student: So that is equal.
Student: Yes, all the sides are equal.
Student: This might want to go in –
Student: Like that to that.
Student: That.
Student: And none of them are parallel, do you see any parallel lines?
Student: They couldn’t be parallel.
Student: (?) that and that out.
Student: These two sides come together as a point.
Student: They meet right here, these two right here meet at a point, these two right here meet at a point, these aren’t straight, so when it says equilateral, doesn’t it mean that all angles are equal?
Student: No, it doesn’t have to be.
Student: It just means all the sides are equal to each other.
Student: Yes, (?)
Student: Does that include (?).
Student: Pull your side up a little more (than that one?)
Student: This side needs to be bigger.
Student: No.
Student: Yes, because that’s too small (?).
Student: So do you have any questions?
Student: I understand it now.
Student: Okay.
Student: Why do you think it’s impossible?
Student: Because if it’s an equilateral and more than three sides, then it should have at least one parallel side, because if any shape is equilateral then it should have parallel sides, but it obviously isn’t.
Student: Well that’s not what we came up with, but –
RJ: I chose the scout exercise because I had two different groups that thought that this activity was impossible. But the two groups that had come up with solutions had very different solutions, both were correct, so the scouts were allowed to go to these groups, get hints, and the group then, in exchange without telling them the answer, had them demonstrate what they had shared with them, and that was proof, that was an assessment for them and an assessment for me, proof that they could then go back and share this information with their group and they could arrive at a solution.
Student:
We didn’t have anything to measure with, so we used his arm.
RJ: The students acted as facilitators when they were given their hands. They asked the probing questions, they had the young man
and the young lady thinking about what they needed to do next. And as I do, when I’m asking them, probing questions or trying to move them to another level, I assist to see if they understand what I’m saying and they did the same thing. They asked the young lady, okay, demonstrate to us what you think we mean.
Student: So they’d all have to be equal length(?) but it would have to be an odd number, so it would be a pentagon, but two sides can be parallel so one has to be at a slanted side.
Student: And these aren’t parallel.
Student: Yeah, so –
Student: Do you understand?
Student: Yeah, I get it now.
Student: A quadrilateral with two pairs of parallel sides and only two sides equal.
Student: Quadrilateral, it’s got four sides, two pairs of parallel sides –
Student: It has only two sides.
Student: Two sides equal.
Student: Wait a second.
Student: Would that be a diamond, a rhombus?
Student: No because all of the sides would – no a rhombus.
Student: Rhombus all the sides are equal.
Student: Yeah a rhombus, all of the sides are equal.
Student: It’s like a turned-over square.
Student: What about a rectangle? Oh, what a second, it says, andonly two sides equal.
Student: So that wouldn’t be (?)
Student: (?) four sides equal.
Student: So it’s four sides –
Student: Courtney, what is this one called?
Courtney: A parallelogram, I think.
Student: Yeah, but that wouldn’t be it, because four sides are equal, right. Would it be another rhombus?
Student: No, that’s a rhombus.
Student: I mean, because the trapezoid has all the sides equal.
Student: You can try it, I don’t (?).
SJ:
The level of communication going on between the students was really
impressive. Students were talking with each other a great deal about their
ideas, there was all kinds of mathematical language being used, there
was a lot of questioning of why and then really pursue it and try and
understand why does this happen? Why is this impossible? How do I know
those two sides are equilateral? How do I know those two sides are parallel?
Can I find any other answers? Students were volunteering alternative solutions.
So the idea of learning to seek for more than one solution and not simply
be content with the first answer you arrive at.
RJ: Okay, you all have done a great job, you’ve had some very good discussion in your groups. We’re going to share these two clues out as a class. Kia, would you come up to the board, draw your shape, label it, name it, and explain to us how your group arrived at that solution.
Student: What we did here was first we started out as a square to see, because the problem is a shape with two sides equal and two different sides parallel, but not equal and the (?) was shaped to have more than four sides. So we started out as a square, and then we thought of a trapezoid, so we made a trapezoid and we had to figure out which ones were equal. So these two are equal and these two are parallel.
RJ: Why did you not stick with a square? You said you started out with a square, so what did you find out about the square that didn’t work?
Student: Because all of the sides are parallel and equal/
RJ: All of the sides were parallel and equal. So then you went to the trapezoid and what did you find out about the trapezoid?
Student: That two side are equal and two sides are parallel.
Student: Wouldn’t you call it an Isosceles trapezoid, because as you would call a triangle Isosceles, because it only has two equal sides? So wouldn’t you call this trapezoid an Isosceles because it only has two equal sides?
RJ: Very good, yes you could and you could make that more specific. Okay, so could you also add in front of that, Isosceles and would you spell it for her, Megan?
Megan: I have no clue how to spell it.
RJ: No clue. Does anyone know? Okay, Courtney?
Courtney: Isosceles.
RJ: Very good. If were having a spelling bee you’d get it.
RJ:
There’s a lot of vocabulary and terminology that comes up in this unit
as kids are doing these activities. There are a lot of shapes they are
looking at, there are properties, there are these words parallel, equilateral,
opposite angles, Isosceles shapes, regular shapes, a lot of terminology
that comes up and rather than just have the teacher give students the
definition we were looking for ways to help students really understand
the definition and internalize them.
RJ: Now we’re going to discuss the second clue which was clue 3. And I need a volunteer to share with me your discussion and your solution to clue 3. Kiana, you want to share?
Kiana: My girlfriend and I thought it was impossible, because a quadrilateral has four sides and the only way you can have two pairs of parallel lines are if all of the sides are equal.
RJ: So Kiana, walk me through what you tried to arrive at that solution if you say it’s impossible. Did you try any shapes at all?
Kiana: Yes.
RJ:
Okay, would you show us the shapes on the board that you tried and
explain to us why they don’t work?
Kiana: Brittany came up with a shape like this (demo) --but our only problem was is that these two lines right here weren’t parallel because they would intersect eventually.
RJ: Okay. Is that the only shape you tried?
Kiana: Yes, and we knew it couldn’t be a square because –
RJ: Would you draw a square up for me?
Kiana: (demo)
RJ: Thanks, now why did your group say it would not be a square?
Kiana: Because we had to have two pairs of quadrilaterals. It didn’t say two lines, it said two pairs.
RJ: Two pairs of?
Kiana: Parallel lines.
RJ: Parallel lines.
Kiana: It has to be a quadrilateral with two pairs of parallel sides.
RJ: And that square does not work?
Kiana: Because a square has four sides that are equal, and we have to have only two sides that are equal.
RJ: Very good.
Then after they have been with partners we share in a full-class summation.
RJ: Any group try other shapes? Did any group try other shapes?
Of what they’ve done in order that we don’t have any students going away with incorrect answers or thinking that something could not have been done, so we try to make sure that everybody goes away with the same information.
Okay, we’re going to start this activity off by having you do a visualization kind of exercise. So what I want you to do is close your eyes, all eyes closed, pencils down, and I want you to visualize a square. Can you see the square? Everybody’s seeing a square? Open your eyes and see if the shape looks like this. Okay? This is your square. Okay? I want you to close your eyes again. You still have that square in mind, I want you to let your vertical side tilt to the left about a 60 degree slide. Do you have it visualized? Okay, open your eyes and see if it looks similar to my shape. Similar to my shape. Okay? I need someone else to share with me some properties of this particular shape that we have now. Marcy.
Marcy: There are two pairs of parallel lines. Well there’s one pair of acute angles and one pair of obtuse angles.
RJ: What’s the position if the acute angles? You want to come up and show them to me so that the class can see what you’re doing? If you just point on the figure and show me what’s acute.
Marcy: That’s an acute angle and that one is an acute angle too.
RJ:
So what is their position? Think about positional words.
Marcy: --
RJ: Can someone help her with position here? A positional word to talk about these two acute angles, Dustin?
Dustin: This would be the top left and that would be the bottom right.
RJ: And the angles are in what position from each other? I’m looking for –
Megan(?): Opposite?
RJ: Opposite! Wow! You knew it all along. All right. What we’re
going to do, we’re going to play a game with sides and angles, with your
partner, and you are going to be visualizing a particular shape and then
you are going to see if you can draw that shape, 
so
when you get with your partner I’m going to give you some material to
work with, I’ll go over instructions again, but you are going to work
with your partner and you are going to be visualizing shapes and then
drawing them. Okay? I’m going to give you a set of yellow carts and orange
carts and a worksheet, but you can’t do anything until I give you everything
and you wait for directions. Okay? Let me go over the directions that
you are going to need to play this game. You have a partner,
you have two sets of cards.
SJ: In the side angle game, the students are given clues on different colored tag board. One set of clues is about sides, one set of clues is about angles. And they are to pick one from each stack, put the clues together and see if they can draw a shape that meets the properties of the side clue and the angle clue.
Student: This shape is an equilateral.
Student: And has exactly two equal angles.
Student: I thought an equilateral had only four equal angles and it says it has to have exactly two equal angles.
Student: Okay, two equal angles wouldn’t work if all the sides were equal.
Student: This shape has no parallel sides in it.
Student: And it has two pairs of opposite angles.
Student: Well, this one is obtuse, okay? It’s about 123, and this one is acute, because it was about 35, 37 degrees. And this one is about 38 degrees.
Student: Have you tried an isosceles triangle?
Student: That didn’t work.
Student: That didn’t work because it only has one pair.
Student: We tried a trapezoid already, right?
Student: Yes, because a trapezoid has parallel lines.
Student: And it says this shape has no parallel lines in it.
Student: And –
Student: Maybe it’s impossible.
Student: Ms. Jones said it might be impossible and we have to explain why.
Student: So it’s impossible –
Student: We tried a scalene and an isosceles triangle.
Student: And if we do an equilateral triangle it will be the same results as before.
Student: And we tried a trapezoid, but that has parallel lines.
Student: Right, so it’s basically impossible, because if the lines aren’t parallel then you can’t get two pairs of –
Student: Opposite angles.
Student: Yeah.
RJ: Okay, how are you guys doing? Let’s read your clues, because you have one that’s impossible so I want to hear you discuss that for me.
Student: We came up with side B and angle B.
Student: Side B says, this shape has no parallel sides in it. Angle B says and it has two pairs of opposite angles. And it’s impossible because in order to have two opposite angles you have to have parallel lines.
RJ: Show me something that’s opposite. We just talked about opposites. Show me what opposite angles would look like.
Student: Side A.
Student: (?) parallelograms.
Student: Okay.
Student: Because you had obtuse angles and acute angles.
RJ: Now which ones were obtuse again?
Student: This, the diagonal one.
Student: And that one.
Student: And these were acute.
RJ: Okay, so opposite angles are in which direction from each other?
Student: Opposite angles.
RJ: But draw a line to connect them for me. Okay, are you getting opposite angles when you use you triangles? Is there any way in which you can get opposite angles?
Students: No.
RJ: So do you need to continue to try with the triangle?
Students: No.
RJ: Okay, what would you try then? Because you tried a trapezoid.
Students: Parallel lines.
RJ: Okay, are there other shapes that you can try, other than the trapezoid?
Student: What about a rhombus?
RJ: Try a rhombus, let’s see.
Student: We can’t do that because it has parallel lines in it.
RJ: Well, why don’t you draw it, since she suggested it, why don’t you draw it anyway. Okay, now tell me, you had told her that it wouldn’t work. So let’s explain why it will not work.
Student: Because it has two pairs of parallel lines and we don’t want any parallel lines. So we could do another four-sided figure so it’s not a triangle (?) working.
RJ: Can you draw another four-sided figure? You’ve only tried, what did you try, the trapezoid, the rhombus, the parallelogram.
Student: (?).
RJ: Okay, let’s try another four-sided figure, can you think of one?
Student: (?).
RJ: Okay. Now remember you don’t want parallel sides. Oh, something like that. Are any of these sides parallel?
Student: No.
RJ: So what’s the other property?
Student: Two pairs of opposite angles?
RJ: Two pairs of opposite angles?
Student: This one’s obtuse. And this one’s (?), so that’s (?).
RJ: Mark your angles and then connect your opposite angles, because your clue only – read the two again.
Student: And it has two pairs of opposite angles.
RJ: Two pairs of opposite angles, it doesn’t say anything about them being how? Do they have to be equal? Okay, you just need two pairs of opposite angles. Where you were getting hung up at was you thought they had to be equal(?) Okay? Look at this one. This is what you, remember how you drew this for me? And you marked your angles. Now these angles have the same number of degrees, same number of degrees in them. They were opposite but the same number of degrees and if you read this clue again, it says and has two pairs of opposite angles. And you all kept reading in equal, somehow, correct? That’s why you didn’t think that these were opposite and that’s why I had you go back and draw those. Do you understand what was throwing you off? Great, okay, you can’t read more into the clue than you already have, all right? You did a great job walking through that. Okay.
RJ:
We’re getting ready to do another activity and this one’s called Animated
Shapes, it is so much fun, I love this one. You are going to use the ribbon
again and remember how we visualized shapes in our mind with our eyes
closed? You’re going to be standing with your group, four people in a
group and you are going to start off with a shape with your ribbon in
that shape and then you’re going to have another clue, and somewhat, you
are going to have to decide who is going to move to make that shape that
the next clue gives you? Okay?
Okay, you’re staring with an equilateral shape. Equilateral means what?
Student: All sides equal.
RJ: All sides equal. So does that meet that? Oh, okay. It looks like a square to me. End up with a shape in which only one pair of sides is parallel.
Student: (?).
Student: Wait, let go of that.
RJ: So help me, where are your parallel sides?
Student: Here.
RJ: One set was parallel. Great! So you went from a square to a (?)
Student: Trapezoid.
RJ: Trapezoid. Good deal. Good deal. All right, you’ve got two more to go.
Student: Hold it right there.
RJ: Excellent, excellent, good way to measure, you use the tile. Okay, stay right there, I’m going to give you your next clue and I don’t want you to move it. End up with a shape in which only one pair of sides is parallel. So talk me through it, tell me what you’re going to do.
Student: We’re going to (?) a trapezoid.
RJ: So how are you going to make that and who is going to move?
Point to the sides when you (?) a parallel. Very good. Isn’t this fun?
Student: Yes.
When we developed Mathscape we had a number of different learning goals in mind for kids. One category were content learning goals and those aligned pretty well with the NCTM standards right now. We tried to really develop Mathscape so that both the mathematical topics grew within a unit and across all three grades. so for instance, if you look at the geometry strand, there are some ideas that get developed in geometry at the 6th grade that are really laying the foundation for other concepts. By the time they get to 7th grade that gets build on as more formal mathematical language and notation gets brought in. So the learning becomes a little more formalized. And by 8th grade we’re really expecting kids to use formal mathematical language in geometry and pull together the concepts and begin to really abstract.
Student: (?) shape, two parallel sides and no sides equal. So that’s how it was.
Student: Okay, but then what do we have to go to?
Student: And then we’ve got to go to something with two parallel sides.
Student: It cannot be (?).
Student: And they can still, they can equal.
Student: And can this change the size? After they come up with the shape.
Student: So this side bent in, this side bent in, yeah, these two would be parallel. These two are (?) parallel. That works.
I think this program can be very rewarding for the teachers, because they get to know their students abilities and ways of thinking about mathematics at such greater depth than they might using other kinds of materials that don’t encourage kids to talk, to communicate their ideas. A teacher up in front of the classroom can be asking a lot of questions, but until you really start to hear how the students are thinking about the mathematics, I don’t think you get as in-depth an understanding about what your students really understand and how they are making sense of the mathematics.
Student: When we were doing the animated shapes, when we were trying to move them around, it was so much easier, because I realized that a trapezoid, if you moved the top up it could easily become a triangle, and it was just, it was a lot easier than I thought it was going to be.
Student: I found out that a lot of the times we think of the ordinary shapes, the square with the four-sides and everything, but really that doesn’t have to be the ordinary shapes, you can kind of explore a little bit and you might turn it this way and that way and it will have a whole different appearance, although it still may have the same name
Student:
I found out that we could use different objects to measure shapes,
like we used with the tiles in our group and we measured out each tile
to see how many we needed to make a square.
RJ: I provide them with a variety of opportunities to think for themselves, to share that knowledge. When they are empowered with that mathematical knowledge, and they are presented problems later on, and when they take higher level math, they are not afraid to tackle that problem, because they have this repertoire of strategies they’ve developed over the years, they are not insecure about their answers and they will share in a classroom and those are real-life applications that they can take to the workplace and that’s what we’re looking for with our young adults. And that’s where we’re trying to get them and Mathscape does a good job.
Student: I learned that even though this is math, which really isn’t my favorite subject, it was still pretty fun.
For further information about the Modeling Middle School Mathematics Project, please contact WWW.MMMproject.org.
Major funding for Modeling Middle School Mathematics is provided by the National Science Foundation.