The students are in groups of four. Each group has chosen a group member to perform these roles: a team captain, resource manager, recorder, and facilitator. The groups have access to the problem (one page per group) and two packets of manipulatives for a group of four. They also have other resources that they can retrieve, including a page of definitions of quadrilaterals. The students begin working on the investigation in different manners. In some of the groups, students work individually for a while. In other groups, students work in pairs, and in still others, the entire group of four is collaborating. For those groups who initially retrieve the definition page, it seemed to influence the order they attacked the problem. The square, rectangle, and rhombus appeared to be the most straightforward for the students. Mathematically, if two of the diagonals form right angles, then at least a pair of sides of the quadrilateral will be equal in length. If the diagonals intersect at the midpoint of both diagonals, then the figure formed will be some parallelogram. In order for two diagonals to form a non-isosceles trapezoid, the following relationships must hold true: If AB is one diagonal and DE is the other diagonal, then trapezoid ADBE is formed only if the diagonals intersect at point P, which is not the midpoint, and AP/PB = DP/PE. This relationship was quite difficult for the students to investigate and conclude. The students did not choose to measure the diagonals with rulers, and therefore did not pick up on the proportional aspects of the diagonals in a non-isosceles trapezoid.
CATHY HUMPHREYS: Are you having trouble like coordinating yourselves as a group? You are all pretty quiet over here.
STUDENT: That’s what I’m seeing.
CATHY HUMPHREYS: That’s what you’re seeing? Okay, so how could you, what could you do? Let me see – where are you, where are you right now?
STUDENT: Figuring out how to make an isosceles…
CATHY HUMPHREYS: Let me hear from somebody else Aaron. So Carmel, where do you think we are?
STUDENT: We were trying to make all the shapes and then we realized that we couldn’t make the trapezoid.
CATHY HUMPHREYS: You cannot make a trapezoid yet? Okay, but you can make the others?
CATHY HUMPHREYS: Are you positive that you have directions for the manufacturer for each of the others that will always work? And there is no other...not only will they always work but there is no other way to make that shape besides what you’ve given? Are you sure about that? So I want to keep you in mind here, you might be, because eventually...because you always want to…you are convincing yourself that you got this and you are all sort of mathematical friends because you are seeing things the same way, but try to be skeptical. Somebody in the group needs to play the role of well, what about this and what about that, so that you make yourself...because it sounds like there is not enough disagreement; and a disagreement is what your group needs to be a really good group.
STUDENT: Why are these holes not lined up?
CATHY HUMPHREYS: That…they aren’t?
STUDENT: Like I have the middle ones lined up and the end two are but…
CATHY HUMPHREYS: Well, that might be sort of like a manufacturing error.
STUDENT: Because I noticed when I line these two up, only the two holes...
CATHY HUMPHREYS: Alright, so is someone willing to take kind of a leadership role? Are you the facilitator Aaron?
STUDENT: No, she is.
CATHY HUMPHREYS: Carmel, so what I’m going to remind you is you are supposed to coordinate your group and you need to make sure everyone has something important to do. So why don’t you take a status check right now; see what everybody is working on and make a plan together. Alright? Okay.
STUDENT: I just finished making this shape with the big one.
STUDENT: Like I said we need to figure out why it works.
STUDENT: Well, I have something. It is because if you keep it all the same distance from the end then it’s going to be one pair of parallel lines and if you keep both of them in the center then there is going to be two pairs of parallel lines. I don’t know if it works with this one though if they are the same lengths. So let me show you. Okay, so here, if you keep them both one point away from the end then this line is parallel to this line, no matter if you go like this or like this weird scissor shape; same thing with this one. Still parallel, still parallel these two.
STUDENT: So long as you keep the same number of points then the two lines are always going to be parallel?
STUDENT: And if you keep them both centered then you have two parallel lines. It's like whether you go like this because these two are parallel and these two are parallel. Um, it works the first way if you keep them both one point away, one point away. And then I think these two are parallel. I have to check, I don’t know how. And these two I think are parallel and these two. Wait, which one am I talking about right now? I confused myself. These two are parallel and these two are not. So...and then if you center it, it’s not going to work. It will work with one but actually it might work with the other two.
STUDENT: It's a rhombus.
STUDENT: Yeah. Okay, so if we keep them centered...
STUDENT: Does a rhombus have two pairs of parallel lines?
STUDENT: Yeah, so if we keep them centered then you get two pairs and if you keep them the same distance, it’s going to be one pair.
STUDENT: Why aren’t they marked off? Okay, so we now know if it is centered then we have two pairs and…
STUDENT: And if you have equal distance from the end then it has one pair parallel.
STUDENT: So then what if it’s centered and we turn it?
STUDENT: It stays the same; they are both still parallel no matter which way you turn it.
STUDENT: Well this…
STUDENT: Oh, that one stays the same, this one doesn’t. This one is weird so you have to keep it at ninety degrees on this one.
STUDENT: So those are only parallel if it’s ninety degrees.
STUDENT: And then if you turn it a bit then it just goes back to…no, I am confusing myself.
STUDENT: She is confusing herself.
STUDENT: No, it’s the same thing here. It just looks weird but it’s still the same.
STUDENT: And if we move it, what happens?
STUDENT: It stays the same, parallel unless you make it not equal distance, that's all.
COMMENTARY BY CATHY HUMPHREYS: Reviewing the videos of group work was truly fascinating for me; it is rare for a teacher to be able to watch students’ conversations so closely.
Throughout the block I saw difficulties with the use of mathematical language and definitions. The frequent use of pronouns (“it”, “they”) obscured what students were trying to communicate. Other examples of difficulties in communicating mathematically include imprecise or incorrect uses of terms (regular, diamond) and definitions.
In Part B an important question was asked but was not taken up with interest by the other group members: “Can a trapezoid be a rectangle?” His team member answered, “I don’t think so,” but did not justify his answer using a definition. This makes me think that he had an intuitive notion that a trapezoid was never a rectangle, but he did not articulate why. I am afraid that unless students are allowed more opportunities to develop the notion of the importance of precision in mathematical language, they will not be able to use definitions to support their ideas.
In Part C as well as in the closing of the lesson, I refer to “mathematical friends.” This notion came from Thinking Mathematically by Burton and Mason, a book about mathematical problem solving in which the authors talk about a hierarchy of certainty when trying to write a convincing argument. Convince yourself (the easiest), convince a [mathematical] friend, and finally, convince a skeptic. Developing a skeptical mindset and not jumping to conclusions too quickly is another hallmark of good mathematical thinking.
It was fascinating to me that Carmel and her group thought it was sufficient to measure the alternate interior angles of one figure in order to prove that the opposite sides parallel for all like figures.
Our class had studied the triangle congruence theorems in December and had been able to show that corresponding parts of congruent triangles are congruent. They also had studied theorems (and their converses) about the angles formed by parallel lines and a transversal. And all year we had been working with the idea of a general argument and the difference between showing something is true for a particular case and showing it is true for all cases. Carmel and her group were able use the knowledge that if alternate interior angles are congruent, then the two lines are parallel. But they were unable to access the need for a general proof and to use triangle congruence to do so.
It keeps hitting me that when students are confronted with a new problem that they have not seen before, it is very often difficult to access skills they have learned in an isolated context or setting. Practicing CPCTC over and over again with different types of problems did not help these students recognize that proving triangles congruent and using this to show that alternate interior angles were congruent, rather than measuring, would accomplish the general proof they needed. Knowing how to use a tool is very different from knowing when to use it, and students need a lot more practice with problems that require them to dig deep to find, from all that they know, the tools and ideas that will help them solve a new and unexpected problem.
COMMENTARY BY COACH DAVID FOSTER: Cathy makes use of these roles throughout the class to get groups started, making sure everyone is participating, retrieving materials and cleaning up and recording findings and communicating their conclusions. The issuing of the materials was quite purposeful. Cathy knows that if the students have one problem they will need to begin by sharing the instructions together. She also handed out two sets of manipulatives so that students could have hands-on experiences, but not so many as that each team member would be off working only by his or her self. Cathy wants to promote individual think time, so how students actually reconvene as a team is dependent on their own learning needs. Most groups were able to use the manipulatives and their understanding of quadrilaterals to determine how the arrangement and size of the diagonals determined the square, rectangle, rhombus and parallelogram. The students found how to form a “geometric kite” rather easily, but struggled with how to described all the different positions and size of the sticks that would make a kite. Students used the holes in the manipulatives as a measuring tool, but very few connected how the holes of the two sets of sticks were proportional. The trapezoids (isosceles and non-isosceles) were the most challenging quadrilateral to find and define for the students. This was especially true of the non-isosceles trapezoid. Although, they were able to construct the trapezoid with the manipulatives, they were unable to ascertain the exact relationship of the diagonals necessary to determine a non-isosceles trapezoid. Realizing the proportional relationship between the intersections was not very obvious to the students. The positioning of the holes (although proportional) did not seem to help students see this important relationship.