curriculum

Estes & Mints 2016

Instructions: A Models Approach

Chpt 13 Fouth Grade Case Study

You Will Know • The opportunities for using diverse instructional models in an elementary classroom • The planning components needed to implement an elementary unit • The relationship between desired objectives and instructional models • The role of student achievement, experiences, and interests in choosing academic content and instruction You Will Understand That • Effective instruction requires careful planning and strong pedagogical knowledge • Planning involves the application of diverse instructional models You Will Be Able To • Reflect and comment on the instructional planning of an elementary teacher • Consider how you would develop a detailed plan for a lesson or unit you might teach • Determine whether the teacher described in this chapter used the principles of backward design V eronica Evans, a fourth grade science and mathematics teacher at Wilson Elementary School, relaxed one weekend before the fall semester began. She was reminiscing about why she had become a teacher. The answer was clear and had been so from the beginning of her teacher preparation—she wanted to be the kind of teacher who made the basic STEM concepts of science, technology, engineering, and math familiar to young students. Lately, she had modified her thinking to give particular consideration to the girls in her class. Studies had been repeatedly reporting that girls in school come quickly to believe that the sciences and mathematics are not for them. Like many folks, Mrs. Evans believed that the nation needed more females in these fields of study and work. Her views were reinforced by various organizations devoted to reversing the anomaly—groups like GEMS, Girls Excelling in Math and Science. WEB RESOURCE Girl Power Visit gemsclub.org or drop a note to [email protected] for more information about Girls Excelling in Math and Science. Worth a look also is a particular comic strip featuring Hispanic characters and one very ambitious young female. Don’t miss the cartoon ahead of the article entitled “Girl Power: The STEM Issue Branches Out to the Funny Pages.”25

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But how to make the basic concepts of math and science appealing to all her students, girls and boys alike—that was the question Mrs. Evans faced with each new crop of young minds that came before her. Her answer was to teach the concepts of math and science in ways potentially appealing because they could be seen as ways of thinking about relationships of all kinds. In her introductory lessons in math, she decided to focus on a particular Common Core State Standard, the mathematic standard for grade four geometric measurements: understand concepts of angle and measure angles. WEB RESOURCE Math Standards Visit the Common Core State Standards website (corestandards.org) and explore the math standards either by grade level (K–12) or domain (including geometry). The standard seemed to lend itself nicely to both a logical and an aesthetic way of viewing the world, features that had potential to appeal to girls and boys alike if presented in a way that had more appeal than formulas. Lines and points are universals in the experiences of everyone. There are yard marker lines on playing fields, lines on basketball courts, lines on the floors and ceilings of buildings—all of which demark edges and boundaries. Though they often do define boundaries, lines also represent infinity. On sailing vessels, one never refers to ropes: everyone knows those are lines. Every family tree is drawn with lines that track the family’s lineage. Straight lines can be assembled to represent figures of three or more sides, and the straight lines at their intersections, called vertices, form angles—sometimes right angles, sometimes acute angles, sometimes obtuse angles. The meanings of those words, aside from their special use in mathematics, would no doubt make for interesting discussion and exploration. As she continued to think, Mrs. Evans considered how the two ends of a line can be brought around curvilineally to form a circle. The circumference and the diameter of a circle are represented by lines of different kinds, curved and straight, and the famous number π is the relationship of the two. The fact that the ratio of the circumference to the diameter of a circle is constant but an indefinite number has been known for so long that it is quite untraceable. What we do know is that it has fascinated mathematicians and other thinkers for centuries, dating back at least 4,000 years ago to Egypt, and is even mentioned in the Bible (1 Kings 7:23) and in popular culture—in the title of a movie, π, in a novel by Carl Sagan in which it is suggested that the creator of the universe buried a secret within the digits of π, in the lyrics of a song, “Pi” included in the album Ariel, by Kate Bush. No single number has held more fascination for humankind. The question for Mrs. Evans and for all teachers of mathematics is whether the topics of mathematics can be taught in ways that provide students with glimpses of such fascination. This day she smiled to herself as she realized that her own enthusiasm could cause her to go off in many directions at once. There was always a limit to what could be included in one unit and how much students could absorb. “Keep in mind how many times the same concept has to be experienced before it can be owned,” she reminded himself. The time had come to order her own thoughts in developing the unit on lines and angles

Mrs. Evans’ Plan Typically, Mrs. Evans planned her units of instruction to last two to three weeks, depend-ing on the content to be covered. Still, every journey begins with small steps, and this plan was no different. Working from the appropriate Common Core mathematics stan-dard, and using this as the basis of designing backwards from the expected outcomes, she specified the general objectives for the first few lessons of the unit: Students Will Know • How to draw and identify lines and angles and how to classify shapes by prop-erties of their lines and angles • How to draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines Students Will Understand That • Lines are the foundation of shapes Students Will Be Able To • Identify linear features of two-dimensional figures • Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size • Recognize right angles as a category and identify right triangles • Recognize a line of symmetry for a two-dimensional figure as a line across the figure, such that the figure can be folded along the line into matching parts • Identify line-symmetric figures and draw lines of symmetry The following lessons are based on the standards and the proposed unit objectives. Lesson One: Words We Use to Talk about Angles Instructional Model Vocabulary acquisition Estimated Time Thirty minutes Rationale for Choice of Model Mrs. Evans believes that words are the basis of concepts, and she wants to make certain that the students understand the basic ideas that will be discussed in the unit. In addition, the words we use to label and discuss concepts often have interesting histories and reflect how we understand phenomena. Vocabulary allows learners to engage in accurate descriptions of concepts and to understand what they hear and read about the concepts they are studying. The first of three lessons Mrs. Evans planned on the topic of angles is aimed at an understanding of the vocabulary used in talking about angles. She starts with a brief pretest of the words point, line, ray, segment, angle, and degree. The pretest is a matching exercise in which students are shown drawings of each of the words and are asked to pick the word that would label each and would form the basis of the following discussion:

Mrs. Evans: These words and their drawings are the basics we will need to talk about the idea of angles. Let’s review them together. What can you say about a point? Robert: It doesn’t go anywhere, it just sits there. Mrs. Evans: That’s right. We say that a point does not have length or width or depth. It actually doesn’t have as much dimension as the drawing we use to represent it. It is more like a place we can only imagine and mark with a dot so we’ll know where it is. We might say, “At what point did you remember the answer?” Notice that when we say that, we are using the word figuratively. There is no actual point, is there? Still, we need to point to it by naming it. Now, what can you say about a line? It contrasts nicely with a point, we might say. But how does it differ from a point? Margie: Does this have anything to do with the way you have drawn it? Mrs. Evans: Yes. You’re on the right track. Margie: Does a line actually go off in both directions? Like the directions of the arrows you have drawn on each end? As if there really were no ends? Mrs. Evans: That’s exactly right, Marge. A line really goes in two directions without ending in either direction. Like to infinity. But like a point, this is what we imagine, since we can really only imagine infinity. Just think of a line like Margie thinks of it—it has no end we can mark. Mrs. Evans continues with the word ray and reminds students that a ray has a specific beginning, but no identified end. She compares that with the definition of a segment. When discussing the line segment, the conversation continues. Marie: Looks like it maybe has two points rather than arrows, one on each end, and doesn’t go farther than the points in either direction. No arrows on the ends. Mrs. Evans: Very good, Marie. Most of the time when we talk about a line, like a “line” of students at the lunch bar or customers at the checkout in a store or cars at a stoplight, we really mean a “line segment.” I know it may sometimes feel that way, but it really doesn’t go on forever in either direction, does it? Let’s look now at the prefix seg-. Like a lot of English, this came to us from Latin and it means “cut.” A segment is a part that is “cut.” Mrs. Evans draws a line, arrows on both ends, and then marks off a segment with two separate hash marks. She mentions that the verb hash actually means “to chop into pieces.” Mrs Evans: What we’ve done here is to chop a segment out of the line. Let’s think about our angle words. What strikes you about the word itself, angle? Marcus: It looks like ankle. Mrs. Evans: Well, to my amazement, the word angle comes from a Latin word meaning “corner” or “bend.” What do you notice about your ankle? Cheri: Your ankle is a bend between your leg and your foot. Is that where it gets its name? Mrs. Evans: Yep, that’s exactly where we get the name. Matter of fact, angle is related to a Greek word that means “elbow.” When we think of angles, we might think of these angles in our arms and legs. Angles are bends, aren’t they? We measure these bends by degrees. What else can you think of that is measured by degrees?

Ronald: How about the temperature? Like when the thermometer says it’s cold outside, like 30 degrees. Or when we get a fever, we say how many degrees it is. Mrs. Evans: Right again. And a degree is a step, or a grade. Each mark on a thermometer is a “step” or “grade.” Those steps differ on two different ther-mometers—the Fahrenheit and the Celsius, or centigrade—scales. The centigrade thermometer is divided into hundredths, each division thought of as 1/100 of a degree between the freezing and boiling points of water. The prefix centi-means “hundred,” as you may know. Now, when we come to measuring angles, I’ll explain how each angle is measured in degrees up to, but not as high as, 360. You may be wondering about where we get that number, 360. For now, I’ll tell you that the way we measure them, angles will always be less than 360 degrees. There is a geometric figure that does measure 360 degrees, but I’ll leave that for you to think about for now. We’ll get back to our discussion of angles tomorrow, when all these words will take on better meanings for you as we use them in talking about angles. Assessment Following this lesson, a variety of common objects related to one or more terms taught in this unit will be placed on a table. These will include things such as a string with knots on each end, a small carpenter’s square, a hammer with a nail pull, a small rod, a hinge, a round coaster for a glass, a hand-operated juicer, a pair of eyeglasses, a boot or high-top shoe, a pair of scissors, a book, a kite, a dollhouse-shaped straight-backed chair, or any fairly small object that can be defined in shape or use with reference to angles. Students will be asked to pick one of the objects and be prepared to “show and tell” how the shape or the use of the object relates to what they know about angles. Lesson Two: Exploring Angles Instructional Model Concept attainment Estimated Time One hour Objective Students will be able to identify the essential characteristics of three common types of angles. Rationale for Choice of Model A concept is a general idea derived from encounters with specific instances. Before this lesson (and the lessons and units that will build on it) can proceed, it is crucial that students have a working definition of angles. As we said in Chapter Four, all concepts have (1) a name and definition, (2) examples of items that are included in the concept class, and (3) critical attributes that define the concept. There will be many opportunities for students to refine their concept of angle in the lessons and learning activities to follow. These and many other very sophisticated concepts will be built on the initial understanding of what an angle is and is not. The second lesson on angles is aimed at understanding what an angle is—identifying the essential attributes and the different types of angles, by examin-ing examples of items that have angular shapes. This is a variation of the concept

attainment model. Mrs. Evans projects clip-art pictures on the screen (she could also have opted to draw the pictures on the board) like the following: A check mark A pyramid, or triangle A circle A u-turn A large X A square A straight line A parallelogram A right triangle Mrs. Evans: So, looking at each of these pictures, which would you say are examples of angles? Notice that some pictures have more than one angle. Maria: That check mark is one angle, isn’t it? Ralph: The circle is not an angle, I think, but the X inside the circle seems to be. Mrs. Evans: Correct, Ralph. And how many angles do you see in the X? The discussion proceeds until all the examples are classified as angles or not angles, remembering that an angle is a bend. Students count the angles in the illus-trations that included more than one angle: the X, the square, the right triangle, the parallelogram, and the triangular pyramid. Next, Mrs. Evans draws three different angles for students to examine: an acute angle, a 90 ̊ angle, and an obtuse angle. She leads the students to see that these three differ in type, related to the idea of 90 degrees. Mrs. Evans: An acute angle is so called because it is sharp, like the angle of a knife’s blade. We refer to excellent vision as acute eyesight, meaning “sharp” eyesight. We might call an angle of less than 90 degrees a sharp angle. By contrast, obtuse means “dull” or “not sharp.” A leaf that has a rounded end is called an obtuse leaf. What do you think a leaf with a sharp end would be called? As an extension of the concept attainment exercise, Mrs. Evans explains the idea that the four 90 ̊ angles in the square add up to 360 degrees. Mrs. Evans: So, you see, all the way around is 360 degrees. That is true whether we’re talking about a square or a circle or this globe (pointing to the classroom globe of the earth). The lines of longitude, which you might call the “up and down” lines, are numbered from zero to 180 to get exactly halfway around. All the way around would be twice 180, or 360. Mrs. Evans shows this on the globe and follows with explanation of how the lines of longitude intersect at 90 degrees at the equator, a special line of latitude. She men-tions that the “equator” is a line dividing the globe (and earth) into two “equal” parts. Mrs. Evans: Lines of longitude are also measured in degrees, as are the lines of latitude. This sidebar discussion is a deliberate attempt to set the students up for the next lesson, concerning how to measure angles with a protractor.

Assessment The assessment of angle recognition is conducted with a matching exercise. The task is to classify a number of pictures of angular objects as acute, right, or obtuse. The pictures are of common objects like a rectangular tabletop, a straight-backed chair, a utility pole, a television antenna, a fishing rod and line, and a drafting T-square. Lesson Three: Measuring Angles Instructional Models Direct instruction and vocabulary acquisition Estimated Time One hour Objective Students will be able to measure angles with the use of a protractor and to draw arcs and circles with the use of a compass. Rationale for Choices of Model By the time of this lesson, the students should be familiar with the basic vocabulary and concept of angles. The new words in this lesson will be protractor and compass, tools that have use in drawing and measuring angles. The goal now is that they become more sophisticated in their understanding of the concept of an angle, including how angles are measured. The direct instruction model is ideal for teaching skills and understand-ings such as these. The model is grounded in what Hattie and Yates (2014) refer to as “ostension,” the act of showing, pointing out, or exhibiting. Direct instruction is char-acterized by ostensible teaching, or teaching and learning by showing and then having students practice the new learning together and independently. As we defined the model in Chapter Three, the process can be expressed as “I do, we do, and you do.” We might also add that ostension is more than showing by the teacher. Direct instruction must involve learners, in part by letting them become doers rather than merely receivers. Mrs. Evan’s presentation begins when she draws a large X on the board to create four angles that share the same vertex at the center, reminding the students of the previ-ous lesson in which they looked at X as a drawing of four angles. Next, she explains that the students will learn how to measure angles with tools made for the purpose. She holds up a large compass and a protractor, explaining that compass means “to go around” or “to encompass with a perimeter, a border or circumference that goes all the way around an area.” She draws a large circle on the board, using the large compass. She then illustrates another meaning of compass by projecting from her electronic tab-let an image of a compass showing direction in degrees from magnetic north. Mrs. Evans: This compass shows all directions: north, east, south, and west. Using north as zero, it tells how many degrees from north it is pointing. Can you guess what the total degrees on the compass are? This reinforces and establishes the idea of 360 degrees as the total of a circle. She next uses the large protractor to show how the size of the angle she has drawn can be measured by placing the midpoint mark on the vertex, aligning the zero line of the protractor on one of the angle’s rays, and reading the degrees on the protractor where the opposite ray intersects. Mrs. Evans: This is much easier to show than to explain. In a minute I’m going to let you practice it yourself. If you have any difficulty, just ask for help. By noting each of the four angles’ measure, she then adds them together to get the total of 360 degrees. She reminds the class that the 360 degrees was what they

found yesterday by adding together four 90 ̊ angles. She also reminds them what they just saw on the magnetic compass. That sum is also the total of degrees in the circumference of any circle. What the students are actually measuring in degrees is the degrees of arc of the circle cut by the angle’s rays. Using the large compass, Mrs. Evans can draw the arc, the part of the circle, on the angles she measured before. Next, the lesson turns to guided practice. Each student has paper and a small ruler for a straight edge, a handheld compass, and a small protractor. Working in small groups, the students practice by drawing two intersecting lines on their paper, creating the four angles of an X. They then measure the degrees of each angle with the protractor; when the four numbers are added together, they total 360, allowing for small error in measurement. When students run into any difficulty, Mrs. Evans is available to provide feed-back and suggestions for correction. Once Mrs. Evans believes that students are pro-gressing toward automaticity, she asks them to work on a similar task independently as she monitors the class. Assessment The day following the direct instruction, the students are given three angles on a sheet of paper and asked to measure them with their protractors. One of the angles is acute (less than 90 degrees), a second one is a right angle (exactly 90 degrees), and the third is obtuse (more than 90 degrees.) They are deliberately constructed so that the total of degrees measurement in the acute and the obtuse angles is 180 degrees. This leads to a discussion of how the measure of the third would, by inference, lead to the measure of the first, and vice-versa. Epilogue The next lesson centers on the topic of supplementary angles, angles whose sum is the measure of a straight line of 180 degrees. This will be found true of each pair of adjoin-ing angles formed by the X the students have seen in each previous lesson. This concept will lead directly to the fourth grade Common Core standard related to recognition of angle measurement as additive. This aligns well with the concept development model. Mrs. Evans also uses a graffiti activity to review the unit’s know objectives. In general, Mrs. Evans is pleased with the unit. One of the motivating factors for her was to teach the unit in such a way that the girls in her class would find the material engag-ing and would develop more confidence and interest in learning math skills. She has not, however, made any explicit efforts in this area. None of the activities or mini-lectures was specifically aimed at girls, and she did not reference any women mathematicians. Now that she has taught the unit, Mrs. Evans will make a concerted effort to make certain that all students can relate to the material she is teaching. In fact, she is thinking about the relationship between success in science and mathematics and social justice issues. That might be a hook that engages all students—across genders, social class, and ethnicities. WEB RESOURCE Khan Academy The design of this particular set of lessons was based in part on brief lessons related to “Angles and Intersecting Lines,” available from Khan Academy: khanacademy.org. Check out the resources available at Kahn, a treasure trove of ideas for teaching many of the subjects and topics in the curriculum. All Khan Academy content is available for free at khanacademy.org. Also, check out possible data sources to use in mathematics lessons at Corps Watch: corpwatch.org. There are articles, cartoons, and data sets about issues that your students might find engaging.

This case study took place in a fourth grade classroom during a math unit on angles. Mrs. Evans, the teacher in the class, has designed a unit with a variety of instruc-tional models—all aligned with the unit’s objectives. The unit began with vocabulary acquisition and included direct instruction, concept attainment, concept development, and a graffiti review. Mrs. Evans felt comfortable with these models and worked hard at being able to implement them successfully. These models are only some of what Mrs. Evans has in her instructional toolkit, but they were chosen for their alignment with the objectives and assessments that were sketched out in the unit. Using a variety of instructional models in any classroom can be difficult. As Mrs. Evans discovered, it takes some time to master these models, and she felt more comfortable with some than with others. However, she was surprised by how well her young students were engaged by the variety of instruction, and she was grateful for the classroom management procedures she had in place in order to support her students’ success. EXTENSIONS ACTIVITIES 265 1. You might have noticed that Mrs. Evan’s lesson plans do not have lesson specific learning objectives. Choose one of the lesson plans and write learning objectives that are aligned with both the assessment and learning experiences of the lesson and the provided unit objectives. 2. Think about a unit you have taught or will be teaching. What are the concepts—the vocabulary—you want students to master during the unit? List the words and think about an appropriate instructional model that will help your students learn these concepts. 3. List the steps you would take in planning a lesson in a unit. What would you do first, second, and so on that would still keep you true to backward design. 4. Draw a flow chart for the decisions that Mrs. Evans made as she planned this unit. 5. Script the directions for a graffiti activity that could be used as a review for the fourth grade unit described in this chapter. REFLECTIVE QUESTIONS 1. Think about the reasons why young girls might turn away from math, science, engineering, and technology in late elementary and middle school. What can you find out about why this happens, and what recommendations can you find that might turn this tide? 2. Skill and knowledge in mathematics can make or break a student’s academic career. Why is this? What is it about algebra, for example, that helps those who are success-ful in algebra graduate from high school? 3. What problems are inherent in using instructional models in the classroom? What difficulties or opportunities might elementary students have as they participate in some of the instructional models presented in this text? 4. How should Mrs. Evans prepare her students for the routines and procedures associated with the different models she is using in the classroom? 5. How would you evaluate the use of backward design by Mrs. Evans? Was it used? How? Was it used appropriately? How do you know?