Just finished the final, and didn’t feel right without posting one last time! I’m all out of words, so thought I’d just resort to a pic of Calvin & Hobbes to show how I’m feeling….
Thanks to everyone who took the time to check out my blog! Three more weeks until next semester starts, hope you guys enjoy the rest of your summer!
It’s kind of ironic to me that as educators, we work so hard to get our students to use key mathematical terms while explaining their solutions; but at the same time, sometimes we have to stop students from using these terms in order to truly evaluate if they understand the concepts and terms being taught.
I was surprised in this article that so many groups didn’t use the wedge as a unit angle, because to me it seemed like the easiest solution. This is pretty surprising, because I rarely think the “easiest” solution is the “right” one. I was amazed at how complex the other groups’ answers were – making a scale from largest, larger, large, large medium, medium, small medium, smaller, to smallest. I think it just reminded me to never assume what students are going to come up! Furthermore, given the students’ responses, I think there were several moments in this activity that could have led to other discussions. It’s something that I think I might struggle with as a teacher- keeping focused at the task on hand. Sometimes a student will say something and I think it brings up another great point, which leads to another, and so on…. and the next thing you know, the lesson is no longer on the unit-angle concept, but instead more on whether circle size is important to angle measurement. I know that the concepts are related, but I fear that I’ll run out of time before getting my students to make the proper correlations.
ANGLE DEFINITION
While degrees are the most commonly used units of angle measure, there are also other units. For example, angles are sometimes measured in radians in order to simplify certain calculations. The radian measure is defined in the International System of Units (SI) as the ratio of arc length to the radius of the circle. For 1 radian, the arc length is equal to radius:
An angle can be defined as the union of two rays with a common endpoint. (A ray begins at a point and extends infinitely in one direction.) The common endpoint is called the vertex (A in the figure below), and the rays are called the sides of the angle.
Angle PQR is an angle whose sides are opposite rays. This type of angle is called a straight angle.
Angle QPT is an angle whose sides (PQ and PT) are coincident. This type of angle is called a zero angle.
ANGLES IN POLYGONS
The sum of the measures of the angles of a triangle is 180 degrees. Notice in this diagram that the diagonal from one vertex of a quadrilateral to the non-adjacent vertex divides the quadrilateral into two triangles:
The sum of the angle measures of these two triangles is 360 degrees, which is also the sum of the measures of the vertex angles of the quadrilateral.
In a regular polygon, the measure of each central angle is equal to the measure of each exterior angle:
Since the central angles total 360 degrees and the exterior angles total 360 degrees, each of these angles is also equal. (For non-regular polygons, these totals are still equal, but the individual angles are not.)
From the angles video and case studies, I saw just how hard it can be for some students to define exactly what an “angle” is. I can’t say that I was really surprised by this, because to be honest, I would probably have a difficult time trying to come up with a justifiable definition as well. It’s funny, we use mathematical terms all of the time, but I think we rarely examine them to the extent that we saw in this video and read about in our case studies.
I also noticed that many children seem to confuse the length of the sides of an angle, or the edges of a shape, to an angle. I think it’s important to make sure your students understand that the actual length of the sides or the length of the edges of a shape have no correlation to do with the actual degree of an angle. I can still draw an obtuse angle, with “little” sides.
Both the video and case studies demonstrated students thinking “out of the box” as well though. For example, in the video, one little girl was trying to make the point that angles don’t have to be created from straight lines. I also thought it was exceptional how she pointed out that a straight line doesn’t need to be drawn vertically or horizontally, but can also be drawn at a slant, diagonally. Just because you draw a diagonal line, doesn’t mean that it’s not straight.
Many of the children also tend to use their hands when describing an angle, and although I think this typical for many students to “talk with their hands”, I also think having students draw their interpretations is sometimes better (as young kids tend to fidget). And as we saw in the video, when they used both of the hands to demonstrate, they didn’t have a hand left to point out the angle they were trying to create!
I also found it interesting that a lot of students are able to identify the bottom angles in a triangle, but forget about the “top” angle. In the case studies, the teacher noted that she thought this was because the bottom angles look more like the examples of angles that students are typically shown. I never thought about that before, but I think it’s really interesting. It’s kind of like our old case study, where students didn’t recognize an obtuse triangle as a triangle because it didn’t look like the isosceles triangle they were used to seeing depicted as a definition of a triangle. I guess that’s why it’s so important that we provide our students with several unique images, and make sure that they understand the definition of the different geometric terms as well.
Sorry for the vast amount of notes, but I never studied the meaning behind these formulas when I was in grade school, so I found this lesson really helpful!
Relationships of the designs above:
Circles and Circumference:
Areas of a Circle:
Transforming a circle into a crude parallelogram:
The scalloped base of the figure is formed by arcs of the circle.
The length of the base is one-half the circle’s circumference, since the entire circumference comprises the scalloped edges that run along the top and bottom of the figure, and exactly half of it appears on each side. The base length is C/2.
Since the circumference is 2 • pi • r, where r is the radius, the base is half of this. The base length is pi • r
As the number of wedges increases, each wedge becomes a nearly vertical piece. The base length becomes closer and closer to a straight line of length pi • r (or half the circumference), while the height is equal to r. The area of such a rectangle is pi • r • r, or pi • r2.
The area formula of a circle is A = pi • r2.
If r is the radius of a circle, then r2 is a square with sides of length r. Examine the circles below. A portion of each circle is covered by a shaded square. We can call each of these squares a radius square.
In each case, it takes a little more than three radius squares to form the circle. If using approximations, it should always take around 3.14 of the squares to cover the circle. The best estimate of the area of any circle in radius squares is somewhere between 3.1 and 3.2, which we know is roughly the value of pi.
Think about a circle with a radius equal to 1 (r = 1). The circumference and the area of this circle are as follows:
C = 2 • 1 • pi = 2 (pi)
A = 12 • pi = pi
Now double the radius to 2 units (r = 2). The circumference and the area of the new circle are as follows:
C = 2 • 2 • pi = 4(pi)
A = 22 • pi = 4 (pi)
The circumference of the new circle doubled, but the area is multiplied by a factor of 4 (the square of the scale factor). You can replace the 1 with any other number, or with a variable r, to see that this relationship will always hold.
Describe Ms. Scrivner’s techniques for letting students explore the relationship between circumference and diameter. What other techniques could you use?
I love the techniques Ms. Scrivner used in this lesson. First, she led a whole class discussion, giving the students the chance to recall what they already knew about the topic. She even referred to the mathematical vocabulary words written on the board and took the time to review them with the class. She had the students visualize what certain terms meant, as they used hand gestures to recall on previous lessons- such as giving a pumpkin a hug to remember what circumference means. She used certain tricks to have students remember the vocabulary as well. For example, there are two i’s in radii, so it must mean two or more “radius”. Once the students reviewed properly what they already knew, she led the class in a new lesson to discover the relationship circles have between their diameter and circumference. The students used centimeter tapes to find circles around the classroom that they could measure individually, and then met back with their group members to discuss their data. Then each group shared one circle and its measurements with the rest of the class, as Ms. Scrivner recorded it on the overhead projector. Students were then allowed to use their calculators to come up with a possible relationship between the circumference and diameter.
Like I said before, I loved this technique because I feel that it really gave the students the chance to discover the relationship on their own, which makes it more meaningful to the student. Here’s another great lesson plan I found online about how to teach students that the circumference divided by the diameter equals pi: intermath.coe.uga.edu/tweb/gcsu-geo-spr06/clewis/Circumference.doc
I wanted to share this lesson in particular with the you because it integrates Language Arts, using the book Sir Cumference and the First Round Table: A Math Adventure by Cindy Neuschwander. I know we just completed our Children’s Literature assignment last week, so I thought this was worth mentioning.
In essence, students in this lesson were learning about the ratio of the circumference to the diameter. Compare how students in this class are learning with how you learned when you were in school.
When I learnt this material in school, I was just given the formula: C/D = 3.14 (or pi). It’s funny, because at the end of the video, Ms. Scrivner even mentions that’s how students used to be taught math. At the time, I found the topic really easy, because all I had to do was remember the formula and plug away. But I realize now, while I was watching the video, that I had completely forgotten what the relationship between the circumference and the diameter of a circle is. I had to Google it! I guess that’s what happens when you remember formulas instead of understanding the concept!
How did Ms. Scrivner have students develop ownership in the mathematical task in this lesson?
Students gained ownership in this lesson because it gave them the chance to discover the solutions for themselves. As the instructor, Ms. Scrivner, led the students on the right track, but she was careful not to just “give away” any answers. Even at the beginning of the lesson, she asked students to look at three words: circle, circumference, and circus, and asked them if they noticed any similarities between these three words. One student said “circ” was included in all of the words, which led the class to try and figure out what “circ” meant. One student suggested that they must all have something to do with being round. Another student linked the word circus having to do with something being round by recalling the term the “ring master”. I loved how each student built upon another peer’s idea, and that the whole class was actively involved in the lesson.
How can student’s understanding be assessed with this task?
I think the biggest way to assess each students’ understanding in this activity is through the class discussion… what does each child have to offer the conversation? I also noticed that Ms. Scrivner continued to walk around the classroom and listen in to each group’s discussion, keeping them on the right track. I also think that you can assess a student by the mistakes they make, because although they may be wrong, that still gives you some insight to how they’re thinking. For example, there was one group that wanted to record the measurements of a student’s head… which led Ms. Scrivner into a whole class discussion on what exactly a circle looks like as a shape.
I found this article online, and wanted to share it with everyone: http://www.scholastic.com/teachers/top-teaching/2013/02/demystifying-measurement
It’s called Demystifying Measurement, and written by Julie Ballew. Her school holds a monthly event called Home & School Connection, and during this session, math coach Cindy Wall supplied parents and students with three different activities to help them understand measurement.
The first activity, Lots of Lines, shows participants that measurements aren’t always as they appear at first (which is why it’s so important to know how to measure properly). In this activity, students are given a curly and straight line. The curly line appears to be shorter at first glance, but actually measures longer in length than the straight line. This reminded me of the Ordering Rectangles activity we needed to complete for module 10- where the slimness or chunkiness of the different rectangles led me to misconceive the actual areas and perimeters.
The second activity, Area vs. Perimeter, is very much like the activity the activity we did with our Tangrams in module 9.
The third activity, Making Measurement Tools, has students actually create a ruler. Ballew writes, “One of the most common struggles in our school is using a ruler correctly. Students consistently measure incorrectly even when the ruler is in their hands, and they often miss test questions involving paper rulers. Cindy addressed this with students and parents by having them make their own rulers in order to understand them better. We always have students make paper clocks to understand time — why aren’t they making rulers? “. What a great point! Several other instructors have commented on the website, stating that they also had their students create rulers and believed their students had benefited from doing so.
For Further Discussion…
The only example I can think of when using a nonstandard measurement might be preferred is while cooking. Ever since I was little, I’ve been trying to duplicate my mom’s recipes and it’s just impossible! My sister was even determined to formally write up all of my mom’s recipes one time. It was so funny to watch! My mom would say “just add a little bit of this”, and before she could throw it in the pot my sister would catch it from my mom’s hand and formally measure it before adding it to the recipe. But while she was measuring that, my mom would sneak a “bit of this” and a “tad of that” into the pot behind her back. It was hysterical to watch! My mom would always tell my sister, “Instead of writing so much, you should be tasting it!”. Now that I’m older and cook everyday, I find that I “cook by taste” as well, but I still can’t duplicate my mom’s recipes!! I guess things just taste better when mom makes it!
The last question on the case studies assignment really got me thinking. I truly felt like although we expect students to progress throughout the grade levels by becoming more efficient, accurate, and precise in their measuring skills, I wouldn’t say that there’s any ideas that no longer warrant discussion at a certain grade level. Here’s what I wrote down, but I’m curious to see what other people responded with…
By comparing the cases from second, third, fourth, and seventh grades to Barbara’s kindergarten (case 12), can we identify ideas that, by the older grades, are understood by the children and no longer warrant discussion. What are some issues that still lie ahead for Barbara’s students to sort out?
It’s hard for me to say which issues warrant further discussion, and which don’t, because as you can see in the cases, every individual child is different. Overall, I would like to say that students understand a more efficient, accurate, and precise way to measure objects the further they advance in grade levels, but I don’t think that means that sometimes a “refresher course” wouldn’t benefit the class as well. For example in case 16, Lisa (a fourth grader) incorrectly measured the chalkboard when she made her marks at the end of the meterstick (approximately 39 ¼ inches), but labeled them 39 inches. In case 14, Les (a third grader), took extra time to realize that the smaller the hand used to measure the object, the bigger the number of measurement became because it would take “more hands” to measure the object. This concept of the larger the unit, the fewer the number of units needed to cover a length, is something that I would still expect some fourth, fifth, sixth, (and so on), graders to still struggle with. As an aspiring teacher, I think it’s important to realize that you really can’t say anything no longer warrants discussion. As Josie proved in case 16, your students can always surprise you. And we’ve seen in many other cases this semester, and speaking from my personal experience as well, that some students might appear that they know the information being taught, but then when you revisit the concept, it seems that you’re right back to square one with them. Hence why it’s called a learning process!
Hmmm…. I don’t remember seeing this article on our list from our class syllabus. I opted not to purchase the NCTM membership this summer, and instead went to the library at the beginning of the course and made copies of the listed articles. But I was able to find a free copy of Measure Up available online: http://web41.its.hawaii.edu/manoa.hawaii.edu/crdg/wp-content/uploads/measureup-2007-05b.pdf
One of the main ideas I took from this article was that “Measurement is often thought of as the determination of size, amount, or degree of something by using an instrument or device marked in standard units. Measure Up, however, uses a broader definition of measurement that includes (1) comparing something with an object of known size; (2) estimating or assessing the extent, quality, value or effect of something; and (3) judging something by comparing it with a certain standard. These three aspects of measurement allow students to explore mathematical structures and develop an understanding of qualitative relationships that offers access to more sophisticated mathematical ideas at earlier grades” (page 452).
I found it interesting that the article said that children often say an object is larger or smaller, but rarely specify how it is larger or smaller. This is something I brought up in my last post, about the Ordering Rectangles activity we just completed. I spoke about how rectangle C had one of the largest perimeters, but also the smallest area out of the set. I loved this, because I thought it would lead to a great class discussion if we should refer to it as a small or large rectangle.
Overall, I was really surprised by the first graders’ work highlighted in this article! I think it’s just amazing that they’re able to make so many different relationships between two different shapes. I also really liked that they were being taught to use letters in math, a concept that I’ve seen even adults still struggle with! Not only did I think this was a great lesson on measurement, but I think it gave students a chance to work on some skills that will benefit them in mathematics as a whole- For instance, being able to answer WHY their solution is thought to be correct, being able to establish relationships, and being able to explain their answers orally and through their written work. I just can’t believe this was a first grade class! Must be something in the waters in Hawaii 🙂
ylGarris 