Diverse learners: A teacher must understand how students differ in their approaches to learning and create instructional opportunities that are adapted to students with diverse backgrounds and exceptionalities (Minnesota Standards of Effective Practice, standard 3).
Friday, December 27, 2013
Assessment: A teacher must understand and be able to use formal and informal assessment strategies to evaluate and ensure the continuous intellectual, social, and physical development of the student (Minnesota Standards of Effective Practice, standard 8).
Assessment is constantly being conducted in the elementary classrooms that I am a part of. Below, a detailed analysis of assessment procedures that I used with a first and third grade student, and background assessment information on these two students and the assessments that are used by the school.
Elementary Reading and Writing Assessments
Reading and Writing Assessments Used within the Classroom and School
· What sorts of reading/writing assessments are conducted school wide?
Kindergarteners undergo a five-word test (nap, kid, log, jet, gum), designed by Words Their Way authors, and if they are able to, a Fountas and Pinnell reading running record. They also are given a letter identification and letter sounds test.
First graders are given the Fountas & Pinnell Where-to-Start Word Test, a Fountas & Pinnell running record, and the Words Their Way Primary Spelling Inventory.
Second through fifth graders take the Where-to-Start Word Test, the Fountas & Pinnell running record and the Words Their Way Primary Spelling Inventory. TheMeasure of Academic Progress (MAP) test is administered two to three times a year, and the Minnesota Comprehensive Assessment (MCA) test is given every spring.
· How is assessment information shared with faculty and parents?
Assessment information is discussed between teachers at weekly team meetings, and with parents twice a year during conferences, and more often with family members as needed.
· What sorts of reading, writing and learning assessments does the teacher engage in? These will be both formal and informal assessments.
The teacher uses the assessments listed above, plus the Dolch sight words tests, and assessment and evaluation of the child’s classroom work.
· How does the teacher use this assessment information? How, if at all, does this vary for different children?
The teacher uses this information to assign children to small instructional reading groups and spelling groups. However, the classroom is made up of many different types of groups. For example, research groups may be composed of children who are interested in a common topic who are at different reading levels.
· How does assessment work outside the classroom (e.g., Reading Recovery)
One of my focal students, Arnaldo, participates in a Reading Recovery small group. I helped the teacher complete the assessments described above. Based on these assessments, the Reading Recovery teacher wrote a detailed reading assessment and plan for Arnaldo.
Focal Students’ Work in the Classroom
Arnaldo
First grader Arnaldo was assessed at the beginning of the school year by the teacher and his reading teacher. He was found to have an enthusiastic and positive attitude. He was functioning below grade level in reading and writing. At that time he was beginning to use the first letter in words to determine unknown words. He knew the sounds of some letters and used them to write words with assistance. Arnaldo was able to identify 40 upper and lower case letters out of 72. He was able to identify only 20 letter sounds. He recognized two Dolch sight words out of 25. On a dictated sentence exercise he scored 6/37.
Arnaldo needs to learn to identify the sounds of the letters of the alphabet, and he needs to learn to blend sounds together to make words on his own. He sometimes confuses the order of the sounds as he writes them, but when he is encouraged to attend he makes fewer writing errors.
The assessments helped the teacher place Arnaldo in an early Letter Name—Alphabetic Stage spelling group, and an “A” reading level. The assessments also helped the reading teacher place Arnaldo in a reading group that could meet his needs. The assessments provide a full picture of Arnaldo’s reading and writing strengths and needs at the beginning of the school year.
Tony
Third grader Tony was assessed in early fall of 2011. His MAP reading test shows that his reading skills declined from the spring of 2011 to the fall of 2011. His word recognition, vocabulary, and comprehension are in the low range. However, his fall Fountas and Pinnell running record brings to light his accuracy in oral reading (99%), yet also shows his slightly limited comprehension. The assessment helped the teacher place Tony in an “N” group for independent reading, and a higher “O” group for reading instruction. The teacher gave Tony the Words Their Way Spelling Inventory on September 1, 2011. He scored in the early syllables and affixes stage. The assessments paint a somewhat complex picture of this boy. There seems to be a differential between his oral, fluent reading ability and his comprehension.
Assessment and Analysis of Focal Students’ Reading and Writing that I Conducted
Arnaldo
Formal Reading Tests
Arnaldo completed two formal reading tests with me: a Dolch sight words test and a Fountas & Pinnell running record. He struggled with the Dolch list, so I transferred the Dolch words onto individual cards, which was easier for him. He scored 12/40. (At the beginning of the school year he knew only two Dolch words.) Arnaldo satisfactorily completed a Level B Fountas and Pinnell running record (accuracy rate 99%). This is a good independent reading level for him. His instructional level should be slightly higher than this (Tompkins, 2010, p. 76), which it is, based on classroom observations. His text-to-text and text-to-world comprehension was satisfactory on the Fountas and Pinnell test. He used some context clues, like looking at the pictures, and checking what letter the word begins with (Tompkins, 2010, p. 245) to double-check the words he was saying. In two months of first grade classroom and reading small group experience, the assessments highlight the improvements Arnaldo is making.
Informal Reading Assessments
Arnaldo participated in two informal reading assessments with me: the “Concepts About Print” (CAP) test (Tompkins, 2010, p. 113), and the “Garfield Elementary Reading Attitude Survey.” On the CAP scoring sheet Arnaldo showed that he was familiar with holding and using books. He understands the left-right direction of print. He had a bit of difficulty with letter/word concepts. It was hard for him to identify a sentence. Still, he demonstrated an understanding of the majority of letter/word concepts assessment items (Johnston, 1997, p. 101). On the Garfield test Arnaldo mostly circled the four-point ‘happiest’ Garfield cartoons. This may reflect his naturally upbeat personality, but it may also reflect his limited comprehension. He did not seem to understand the difference between reading at home and reading at school even after we talked about it.
Formal Writing Tests
Arnaldo’s writing was assessed using two formal writing tests: the Words Their Way (WTW) Spelling Inventory Feature Guide and the 6+1 Traits. On the WTW test it is evident that Arnaldo is still struggling to hear short vowel sounds. He is hearing the short ‘a’ and ‘o’ sound. When I encouraged him to break the word down he heard individual sounds more readily. From this assessment it appears that Arnaldo is still in the early Letter Name-Alphabetic stage. I used a piece of classroom writing from September 27 and applied the 6+1 Traits assessment. On ideas and content I gave Arnaldo a 1, because he was just beginning to think about what he wanted to talk about (“I need more time to think.” Culham, 2003, p. 266). On word choice I gave him a 3 because his words were correct but not descriptive. On conventions he received a 1 because he does not yet have control of this area. On organization he received a 1 because the writing piece is not yet shaped. I gave him a 3 on voice because his personality comes through a bit. On sentence fluency he received a 1 because of his very beginning level.
Facilitating Arnaldo’s Progress
Arnaldo benefits from small group and one-to-one reading and writing assistance based on the following formula: read to, read with, talk to, write with. The Cunningham chapter in Gambrell (2007) contains additional phonological awareness activities that I will teach in the small group setting, for example, the “making words” game that splits the word ‘cartoon’ into many small words that students come up with. Further, I can assist him with Words Their Way sorts in the early Letter Name-Alphabetic stage (Bear et al., 2008, pp. 129-150) that compare short vowel word families and examine CVC families.
Tony
Formal Reading Tests
Tony completed two formal reading tests with me: a Dolch sight words test and a Fountas & Pinnell running record. He executed the Dolch third grade test flawlessly. He found these words to be “immediately recognizable,” according to Pearson et al. in Gambrell (2007, p. 46), rather than “arduously analyzable,” a skill that should free him to understand more of what he is reading. His accuracy rate was 99% on his Fountas and Pinnell running record. Tony is proud of his fluent reading ability and his decoding skills. Within the text, his comprehension and recall of details is very good. Tony’s text-to-text understanding, an area that requires higher-level thinking according to Tompkins (2010, p. 261) was good. But when Tony is asked to think beyond the text, to draw parallels between what he has read and his life experiences, he loses interest, and his score reflects that.
Informal Reading Assessments
Informal reading assessments that Tony did with me include the “Motivation to Read Profile,” and the “Garfield Elementary Reading Attitude Survey.” Tony’s “Motivation to Read Profile” is revealing. He consistently rates himself as a pretty good reader, but does not generally give the activity of reading high marks. This survey appears to accurately assess his feelings about reading. When Tony took the “Garfield” survey he frequently sighed about how long it was. On this measure his recreational reading score is only slightly above his academic reading score. Tony did not come close to a total score of 80; rather his score was 49/80.
Intrinsic motivation seems to be key for Tony (Tompkins, 2010, p. 277). Tony seems to be boy who would benefit from being given more choice in his reading and writing assignments.
“Students want to have a say in which books they read and which topics they write about. By making choices, students develop more responsibility for their work and ownership of their accomplishments” (Tompkins, 2010, p. 279).
Formal Writing Assessments
Formal writing assessments that I did with Tony included the 6+1 Traits rubric and theWords Their Way spelling assessment. On the 6+1 Traits assessment, I assessed Tony to be a 1 in ideas/content; a 5 in word choice; a 3 in conventions; a 4 in organization (“The middle makes sense, but it plods along” Culham, 2003, p. 267). I gave him a 3 in Voice, and a 3 in sentence fluency. His low ideas/content score reflects the lack of details and information. As a third grader he could have written more. However, I gave him a 5 in word choice and a 4 organization because of his specific vocabulary, and the clear flow of his piece and the sequential illustrations that accompany his writing. On the Words Their Way spelling test, Tony scored a little higher in early November than he did in early September. Of the final ten words in the Syllables and Affixes stage, Tony spelled three out of seven correctly. Because he spelled some of these correctly, he could probably handle the challenge of the middle level of this stage, rather than the early level. He appears to be crossing into the Syllable-Junctures phase of the Syllables and Affixes stage (Bear et al., 2007, p. 34). Tony has strengths as a speller.
Facilitating Tony’s Progress
I believe Tony is in need of a large dose of academic, non-fiction reading. When Tony and I have explored other topics, such as the Greek gods and goddesses, he utters humorous and curious insights, and readily writes and draws about Zeus and Athena, for example. I will help Tony with a Montessori research project of his choosing on rocks and minerals. This is the perfect medium for reading interesting non-fiction material, developing vocabulary, creating idea webs, outlining main areas of inquiry, and organizing, writing and illustrating a research report.
Bibliography
Bear, D., Invernizzi, M., Templeton, S., & Johnston, F. (2007). Words their way: Word study for phonics, vocabulary and spelling instruction. Upper Saddle River, NJ: Pearson.
Cullham, R. (2003). 6+1 traits of writing: The complete guide grade 3 and up. NY: Scholastic Professional Books.
Gambrel, L., Morrow, L. M., Pressley, M. (2007). Best practices in literacy instruction. New York: Guilford Press.
Johnston, P. (1997). “Concepts about Print,” Knowing literacy: Constructive literacy assessment.Portland, ME: Stenhouse.
Tompkins, G. (2010). Literacy for the 21st century: a balanced approach. Boston: Allyn and Bacon.
Reflection and professional development: A teacher must be a reflective practitioner who continually evaluates the effects of choices and actions on others, including students, parents, and other professionals in the learning community, and who actively seeks out opportunities for professional growth (Minnesota Standards of Effective Practice, standard 9).
Education Philosophy Paper:
A Perennialist Approach Helps More Young Students Succeed
Perennialism: An Introduction
I approach education as an experienced professional who has been involved in public schools in two primary ways: on behalf of my children who just finished attending public schools, and as a person with prior teaching experience. My teaching philosophy has been shaped by experience with the Montessori program that my children attended--the same place where I have recently worked. My philosophy has also been influenced by teaching second grade at an international school for two years where I observed children at all levels of ability desire deep understanding of such subjects as geology, physics, animal classification, and cultural studies. My second grade students were philosophical during literature discussions; they were capable of profound, human insights. These experiences cemented my dedication to thoroughly knowing the subject matter I am teaching so that I can have substantive conversations and activities with students at every juncture. By extension, I have grown desirous of an academic scope and sequence that stresses rich content knowledge, because children’s desire for information is insatiable. A content-rich curriculum must be paired with hands-on activities that the child has chosen, however, or the teaching becomes a rote exercise, and student learning becomes stuffing.
Perennialism fits my desire to offer students a bountiful and relevant engagement with subject matter presented in a logical, sequential order. Perennialism, chocked full of information, but offering a variety of viewpoints, is the best method for assuring that all school children develop their inherent capabilities to the fullest extent possible. Below, past and present views and examples of perennialism, an exploration of the Core Knowledge curriculum, and current efforts to join direct reading instruction with prescribed content for our youngest struggling readers.
Perennialism Today: The Negatives and Positives
Perennialism is a part of the behaviorist educational psychological theory, and is often represented as a top-down form of pedagogy where the rewarding and punishing teacher is in control (Ryan & Cooper, 1998). Perennialism has been described as an inflexible, unchanging doctrine that regards the student as a passive recipient to whom lectures and drills are delivered. Wiles and Bondi (2007) define perennialism as the most traditional and structured educational philosophy.
Yet if perennialism qualifies as the most conservative of the educational approaches, Mortimer Adler characterizes perennialism as “...conservative as in conserving the best of the past” (Ryan & Cooper, p. 311). Adler in his Paideia Proposal (1982) breathed life into perennialism by espousing that all children, not just fortunate children, should be exposed to a high-quality, classical education because it fosters greater equality. Adler felt that it was imperative for students to embark on self-discovery and self-improvement by gaining an understanding of the framework of our government and our founding principles. Adler believed that this type of learning would engender and invite students to demand protection of their civil rights, the right to pursue happiness, and the right to expect a decent standard of living.
Perennialists contend that there is an ordered body of knowledge that students need to know so that the public might unite around a shared identity (Gaudelli, 2002). Having said that, Americans resist group stereotypes and pigeon-holing, and draw attention to the many exceptions to ‘one size fits all.’ Yet, education leader Albert Shanker noted, “The claims of...separatists reflect the attitude that no one group may make a judgement on any other, since all depends on your point of view. This extremely relativistic viewpoint conflicts with the need that all societies have to establish some basic values, guidelines, and beliefs” (Shanker, 2003). An enlightened perennialist educational philosophy outlines fundamental knowledge that forms the basis of education so that children can learn and go on to accept, reject, annotate, extend and embark on the path of their choice. The acquisition of a broad, liberal base of knowledge enables all children to go on to learn more. A primary, fundamental course of study becomes enabling knowledge that ensures the child’s future growth.
Perennialism and E.D. Hirsch
The educator E.D. Hirsch echoes the views of perennialism. Hirsch and his colleagues advanced a specific, sequenced curriculum in literature, mathematics, history, the arts, and sciences—the Core Knowledge curriculum—that provides children with enriched knowledge, expands their vocabularies, and lays an academic foundation on which to build. The Core Knowledge program comprises about 60 percent of the curriculum at each grade level, leaving teachers and students free to pursue additional topics throughout the year. What content does Core Knowledge include? Hirsch poses the question: “Is [the] information often taken for granted in talk and writing addressed to a general literate audience?” (Hirsch, 2006, p. 123). For example, Hirsch believes that young grade school children should be taught about the kings and queens of England so they can then be taught about what spurred the birth of our nation. Children need to learn about autocracy so they can find out about democracy.
Hirsch and Reading Comprehension
Hirsch is a promoter of better reading ability among all students, and struggling students in particular in his recent writings. In his book Cultural Literacy (1988), Hirsch maintains that reading comprehension depends on specific background knowledge, and that struggling students, many poor students, and others lag behind in reading because they don’t know enough essential literature, history, arts, civics, and sciences. Reading ability is the crux of the matter, because it corresponds with learning and communication ability. A child’s reading ability in second grade predicts that child’s academic performance in his or her high school years (Hirsch, 2006). Hirsch feels that reading ability is central to our identity as a country, because our democracy revolves around information and our ability to broadly and insightfully discern the plethora of news and details that we have at our fingertips. But to have a well-rounded reading ability you need to know about a lot of things.
Prior Knowledge Helps Students Perform Better Academically
Cognitive science tells us that the more informed you become about a topic, the easier it is for you to learn more about it (Recht & Leslie, 1988). Recht and Leslie investigated how prior knowledge affected low and high ability middle schoolers’ memory and retention in reading. When a reader already had an understanding about the subject being studied, comprehension was better and there were fewer memory errors. Poor readers were significantly less likely to be able to recall unfamiliar text. The researchers discovered that knowledge of a subject area determined the amount and quality of information recollected, and that prior knowledge helped poor readers compensate for their low reading ability.
Building Young Students’ Knowledge and Skills as Readers
Broad liberal arts knowledge builds reading comprehension and satisfies students’ thirst for learning. Student intellectual capacity is fortified by an ordered curriculum rich in history, science, literature and the arts. Current approaches that depend chiefly on reading strategies such as making text-to-text or text-to-world connections don’t allow class discussions to go far enough. It is time to turn away from the current peripatetic literacy block in elementary school and replace it with a rich, sequential exploration of non-fiction, historical facts, and high-quality readings, even for the youngest readers, paired with choice and hands-on activities. Programs such as these have been shown to increase academic achievement, especially for ELL students and students with special needs.
The Pilot
Ten New York City high-poverty public schools implemented the pilot of the Core Knowledge K-2 Language Arts Program and reaped positive results from 2008-2011 (Dubin, 2012). Administrators and teachers found that their balanced literacy block relied too much on reading comprehension strategies, such as identifying the audience and finding the text’s main idea. Writer’s Workshop emphasized the process of writing but not enough attention was being paid to what was written about. Weak student writing resulted. The children didn’t have enough foundational, topical information to write about. School administrators and teachers became excited after attending a Core Knowledge conference because the curriculum offered students more substance. They also desired a program that paired content knowledge with teaching beginning readers to read, and starting in 2008, the Core Knowledge Language Arts Program, which included both of these components, became available for trial.
Core Knowledge Language Arts Program for Grades K-2
In the Core Knowledge Language Arts Program for young readers, a skills strand teaches decoding, phonics, and fluency coupled with writing (encoding), plus grammar and spelling. The listening and learning content strand sequentially build the child’s knowledge and listening understanding that leads to better reading comprehension down the road (Hirsch, 2010-2011). It is the listening and learning strand that I will describe in depth.
In the listening and learning part of the Core Knowledge Language Arts Program, to build broad knowledge in the early grades, students are exposed to texts in literature, science, and social studies that grow in complexity over time (See Appendix A). The Core Language Arts Program contains more non-fiction texts compared to other traditional language arts curriculums. In kindergarten the texts begin with just a few sentences, but as the year progresses and children’s attention spans lengthen, read-alouds reach two to four pages. Some teachers at the ten pilot testing schools initially wondered if their young students could sit still and listen. However, teachers observed that even kindergarteners and first graders were mesmerized by the rich content.
The literature domain emphasizes fables, stories and myths because Hirsch sees these as the building blocks of later literary and cultural understanding. Multicultural selections are included. For example, the literature sequence features the Caldecott-winning Lon Po Po, the scary ancient Chinese version of Little Red Riding Hood retold by Ed Young. The social studies sequence features Many Nations: An Alphabet of Native America, by Joseph Bruchac. The social studies course of inquiry is ordered so that domains build on each other--Native American studies precede Colonial studies, after which Westward Expansion is explored. Students first learn that as American colonists were fighting for freedom against the British, they themselves kept African slaves. The domain then studies President Lincoln and his battle to end slavery. Similarly, science knowledge develops from examining the five senses in kindergarten to studying the human body in grade one. (Many of these topics are revisited in great depth in later grades.) There are 12 literature, social studies, and science domains per grade level, allowing for two weeks of study and related activities for each topic. Every lesson includes core vocabulary and read-aloud objectives that remind students what they have already heard and learned, and draws out connections to their own lives. Discussion guides provide reflective questions at different levels of complexity that children can respond to verbally, in writing, or by drawing.
The Link Between Oral Comprehension and Reading Comprehension
Read-alouds and discussion are vital to the Core Knowledge Language Arts Program. A child’s reading comprehension is not as sophisticated as their listening comprehension until they are 13 or 14 years old. Even at age nine when children are becoming more advanced readers, listening understanding still surpasses reading understanding (Hirsch, 2010-2011). Oral language competence helps to predict later reading and school success for elementary-age children (Biemiller, 2003):
• By grade 3, children with advanced vocabulary skills possess comprehension abilities at the fourth grade level or higher, while third graders with lower-level vocabulary skills comprehend at the second grade level or below.
• Fourth graders who begin the school year with vocabulary deficits show growing problems in comprehension.
Increasing older elementary students’ academic achievement and avoiding the 4th grade slump depends on enriching the students’ oral language and vocabulary in the early elementary years through a content-rich curriculum.
Outomes
Compared to ten comparison schools in the New York City public system that used other curriculums, the ten experimental schools scored higher on reading comprehension, science, and social studies, while also learning to decode, read and write (Dubin, 2012). Young students succeeded in learning to read and learning to know simultaneously. While other good reading programs, such as the Fountas and Pinnell’s series, teach children the speech to print connection through phonics instruction, the Core Knowledge Language Arts Program shows that ample content can be included in the scope and sequence for young students. Academic achievement increases were especially notable for students with special needs.
Although Core Knowledge instructs what to teach, teachers at the ten pilot schools devised howthey would teach it, using supportive materials from the Core Knowledge Language Arts Program and materials that they gathered on their own. Teachers noted that they no longer had to hunt high and low for leveled instructional materials on topics, but instead could focus on designing small group, hands-on, and differentiated activities. Additionally, the teachers at the ten pilot sites recommended assessment procedures that Core Knowledge integrated into the final Core Knowledge Language Arts Program curriculum.
Bibliography
Adler, M. (1982). The paideia proposal: An educational manifesto. New York: Macmillan Publishing Co.
Biemiller, A. (2003). Oral comprehension sets the ceiling on reading comprehension. American Educator, Spring, 27(1).
Dubin, J. (2012). More than words: An early grades reading program builds skills and knowledge. American Educator, Fall, 36(3), 34-40.
Gaudelli, W. (2002). U.S. kids don’t know U.S. history: The NAEP study, perspectives, and presuppositions. Social Studies, September-October, 197-201.
Hirsch, E.D. (2010-2011). Beyond comprehension. American Educator, Winter, (34)4, 30-43.
Hirsch, E.D. (2006). The knowledge deficit. New York: Houghton Mifflin.
Hirsch, E.D., Kett, J., Trefil, J. (1993). The dictionary of cultural literacy (2nd ed.). Boston: Houghton Mifflin.
Hirsch, E.D. (1988). Cultural literacy: What every American needs to know. New York: Houghton Mifflin.
Recht, D., & Leslie, L. (1988). Effect of prior knowledge on good and poor readers’ memory of text. Journal of Educational Psychology, 80(1), 16-20.
Ryan, K., & Cooper, J. (1998). Those who can, teach (8th ed.). Boston: Houghton Mifflin Co.
Shanker, A. (2003). The importance of civic education. In J. Leming, L. Ellington, and K. Porter, (eds.), Where Did Social Studies Go Wrong? p. 80. Available at http://www.edexcellence.net/doc/contrariansFull.pdf.
Wiles, J., & Bondi, J. (2007). Curriculum development: A guide to practice. Upper Saddle River, NJ: Pearson Merrill Prentice Hall.
Instructional strategies: A teacher must understand and use a variety of instructional strategies to encourage student development of critical thinking, problem solving, and performance skills (Minnesota Standards of Effective Practice, standard 4).
Math Reflection Paper
Part 1
The 4th/5th Grade Classroom Environment and Students
Exploring the Context for Learning
The School
This public Montessori magnet school serves an population of city children. The school serves half white and half children of color. Half of the students receive free or reduced lunch. A quarter of the students in the school are African American students whose home language is not English. About a quarter of the student body is composed of East African immigrant children. One quarter of the children in the school receive ELL instruction. Eight percent of children in the school receive special education services.
Observations of Teaching
The Classroom
This is a packed grade 4-5 classroom made up of 34 students. Montessori materials crowd bookshelves that line every wall. The teacher conducts small group lessons at a low Montessori platform while the rest of the students work independently on Montessori work. The use of Montessori math materials for the many English language learners helps them acquire math ideas more easily because much of the work is done with physical objects and Montessori materials that help to convey concepts in a hands-on fashion. The small group milieu allows for ample discussion time of children’s math thought processes and questions. At this public Montessori school some classrooms are almost pure Montessori, and some are not. This class is a traditional Montessori room. The teacher is devoted to the Montessori philosophy. But to meet the 4th and 5th grade Minnesota math standards and help children perform on the state MCA test she sometimes supplements the curriculum with Math Investigations and Everyday Math material.
There is a two-hour block devoted to Montessori math daily. As a non-certified Montessori teacher, I cannot teach Montessori lessons, but I can learn the lesson and conduct follow-up assistance and guidance with the students. I can also augment Montessori lessons with some material from Everyday Math and Math Investigations as deemed appropriate by the teacher.
Problem Solving
The Montessori math curriculum for 4th and 5th graders contains many aspects of the problem-solving philosophy. Students focus on ideas and sense making as they explore Montessori metal and wooden geometric shapes and extend past knowledge when they apply it to new lessons they are learning. Because students work independently at their own pace on geometry lessons, and because many geometry assignments have an open-ended discovery format, students develop confidence that they can do math and that it can make sense on their terms. Student-teacher small group lessons provide time for the teacher to double-check student thinking and make corrections. Students’ context for much of their math learning—the Montessori geometric inset materials—is familiar and accessible to the students, and provides points of entry for different types of students. The Montessori method requires students to reason, make connections and physical representations, communicate their findings to other students, and come up with solutions to problems.
One geometry lesson that I observed for 4th grade students was entitled, “The Equivalency of a Rhombus to a Rectangle.” (See Appendix A notes.) In this work a red rectangle and a red rhombus are inset next to each other in a flat green metal frame. The red rhombus is cut up and contains an equilateral triangle and two scalene right triangles. In the lesson the teacher guided the students to ask questions and make observations about the relationship between the two figures. Students were invited to take turns moving the shapes around and comparing how they fit into the rectangle and rhombus. Later, students were to assemble a booklet with cut-out paper rectangles and rhombuses, and their written observations about the similarities of the two shapes and the relationship between the two shapes.
Student observations and findings included:
“The small equilateral triangle fits into the rectangle in the middle. The two scalene right triangles can be fit in on either side of the equilateral triangle.”
“Hey! The space inside both shapes is the same!”
(Teacher) “What do we call that inside space?”
“The AREA!”
(Teacher) “The area of the rhombus is equivalent to the area of the rectangle.”
“The base of the triangle is the same as the base of the rectangle.”
“The height of the triangle is the same as the height of the rhombus is the same as the height of the rectangle.”
“The diagonal of the triangle bisects the rhombus in half.”
(Teacher) “Talk to me more about the bisector.”
“It has to be a straight line! It has to divide the rhombus in half, into two equal halves!”
In “The Equivalency of a Rhombus to a Rectangle” student-teacher discussions I witnessed the beginnings of Level 3 Deductive reasoning according to Van Hiele’s geometric thought construct. The 4th graders were going beyond merely the properties of shapes. However, they were at a level of intuition primarily, but beginning to develop some logical arguments to support and prove their observations, e.g., (in the rhombus and rectangle) “The height is the same, and the base is the same. Hey! They have the same area even though they look different!” The 4th grade students in this classroom are engaged in discovering geometric relationships that will lead to later theorems and proofs.
Part 2, Math Instruction
Polygon Capture Lesson Planning Outline
Student description
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Simon is a 4th grader who struggles in school. He does not have an IEP or special education designation. Simon benefits from a scaffolded lesson that provides plenty of time to review prior knowledge and warm-up with practice.
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Purpose
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The purpose of the game is to encourage the student to look at the relationships between geometric properties.
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Objectives
Student will be able to:
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· Describe, classify, and understand relationships among two-dimensional objects using their defining properties;
· Make beginning “If-then” statements about polygons. For example, “If a quadrilateral has these _______ properties, then it must be a __________.” (Level 2, Informal Deduction) (Van de Walle et al., 2010).
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NCTM Standards
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· Classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes;
· Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions;
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MN Standards
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· 4.3.1.1: Describe, classify and sketch triangles, including equilateral, right, obtuse and acute angles.
· 4.3.1.2: Describe, classify and draw quadrilaterals, including squares, rectangles, trapezoids, rhombuses, parallelograms and kites. Recognize quadrilaterals in various contexts.
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Materials
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· Everyday Mathematics Student Reference Book Grade 5 Polygon Capture directions, p. 328, Appendix B
· Polygon Capture pieces
· Polygon Capture property cards
· “Geometry and Constructions” reference section from Everyday Mathematics
· Scissors
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Before Phase
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With Simon:
· Look over the definitions of “polygons” in the Geometry and Constructions reference section of Everyday Mathematics
· Discuss convex, nonconvex, and regular polygons.
· Go over prefixes that are added to –gon that tell the number of sides the polygon has, e.g., hexagon.
· Refer to the tree diagram in Everyday Mathematics that shows how different Quadrangles are related.
· Talk about quadrangles that are and are not parallelograms.
· Assess Simon’s familiarity with the terms parallel, perpendicular, acute, obtuse, right angles. Ask him to illustrate these terms.
· Cut out Polygon Capture pieces and Polygon Capture Property cards.
· Match property cards with polygon capture pieces as a “warm-up” to tomorrow’s polygon capture game.
· As Simon matches cards to shapes, ask him why he thinks his solution is correct and reasonable. Ask him why he did it that way (Van De Walle et al., 2010, p. 60).
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During Phase
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· One of the main goals of this phase is to again probe Simon’s thinking and asking him to explain what he is doing as he plays polygon capture.
· Look for Level I Analysis—an ability to talk about all rectangles, for example.
· Look for and encourage Level II Informal Deduction—if a shape has sides that are not parallel then it is not in the parallelogram family, for example.
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After Phase
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· Show Simon his observations that you have written down.
· Ask him what he notices about his thinking.
· Refer back to the tree diagram of Quadrilaterals in order to discuss relationships between shapes.
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Assessment
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· Note Simon’s level of thinking. Is he at Level I or Level II?
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Reflection About Teaching Math Lesson
Before Phase
I explained to Simon that we would eventually be playing the polygon capture game and showed him the game. I said that we would first do some geometry review. It was very helpful to review with Simon key 4th grade geometry math terms: polygon; vertex; parallel; perpendicular; obtuse, acute and right angles; convex, concave and regular polygons; and the prefixes that are often attached to polygon shapes (e.g., hexagon, octagon). This was done using clear and valuableEveryday Math reference material (See Appendix C) and drawings that Simon and I made.
During our discussion Simon re-discovered that a true polygon cannot have curved sides or open sides, and that the points have to meet (vertices). He noted that, “Parallel lines never meet. They keep going.” He drew parallel lines. He drew and identified acute, obtuse and right angles. He remembered that a quadrilateral has four sides. He enjoyed pointing out that a hexagon was related to a hexagonal prism, and this led to a deeper discussion about the square area of flat shapes (2-D), and the cubic area of three-dimensional shapes. As we held this discussion Simon helped me cut out the polygon capture pieces.
The next day we continued our warm-up discussion. We examined an Everyday Mathematics tree diagram of Quadrangles (Quadrilaterals) and discussed these terms as referring to shapes with four angles and these same shapes have four sides. “How are polygons and quadrilaterals similar and different?” I asked Simon. He struggled but eventually remembered that, “A polygon can have many sides, but a quadrilateral has just four sides.” I urged him with prompts to make a similar if-then statement, and he eventually came up with, “If a polygon has four sides, then it is a quadrilateral.” He noted that, “A square and a rhombus are similar because they have two sets of parallel lines, and they are quadrilaterals. They are different because their angles are different.” At first it was hard for him to see that they also have in common four sides that are all the same length.
During Phase and After Phase
I explained the Polygon Capture game to Simon and asked him if he would like to play it. He declined, saying that he would rather match the different cards with the shapes. I asked him to choose to begin with either side cards or angle cards, and he chose angle cards. He chose to work with a limited number of shape cards (about nine). He chose the card “At least one angle greater than 90 degrees.” He was proud to immediately choose a hexagon and declare, “It has obtuse angles that are bigger than 90 degrees.” He chose the card, “All angles are right angles,” and matched this card with two rectangles and two squares. I drew a scalene right triangle and asked if this could be matched with the card, and he said, “No. It has only one right angle. It has only three sides. If it had another 90 degree angle it would start turning into a square.” He was proud of his statement and his deeper understanding. Our discussion of right angles and rectangles and squares led to a further discussion and examination of the tree diagram of Quadrangles and the parallelograms, rectangles, rhombuses and squares. We attempted an if-then statement about the square and its relationship to rhombuses, and its relationship with rectangles, but this was difficult. He noted again, “The square and rhombus are related because they have four sides that are all the same length!” Then Simon chose the card, “Only one pair of sides parallel” and matched it with the trapezoid. With prompts and references to the Quadrangle tree diagram he was able to say, “If a shape has only two pair of sides parallel like a trapezoid, it is not in the parallelogram group.”
Levels of Geometric Thought
Simon is definitely at Level 1 Analysis as he is usually able to list all the properties of different shapes, but he has a harder time consistently seeing that shapes can be subclasses of one another (Level 2, Informal Deduction).
Simon made several “if-then” statements with prompts. He is moving toward Level 2 thinking.
Lesson Plan Changes
It seemed constructive to drop the game-playing plan and opt for the card matching idea that Simon suggested. The more Simon is able to choose math activities, the more his math anxiety goes down. The more he is able to discuss his thinking, the more he appears to learn. If I had time I would extend this lesson to playing the Polygon Capture game eventually, preferably including another student, too.
What Have I Learned?
Students benefit from having lessons broken down into smaller and smaller pieces. I should have had an assortment of pattern blocks so that Simon could physically play with and handle these pleasing shapes in addition to the paper shapes. For whole class instruction after an all-class lesson I would break the class into smaller groups and have differentiated small group activities prepared that would allow for discussion, comparison, give-and-take. I would spend most of my time with the group that needed the most help. I would rate the groups for their on-task behavior and quiet voices and reward them using our class reward system (e.g., Teddy bear tickets that kids can redeem). I am feeling more and more comfortable with geometry after taking an excellent geometry and measurement course, in addition to this course. I can see that it is vital that the teacher possess a deep understanding of math content so that she can see what her students are trying to get at, and is able to lead them forward based on their glimmers of understanding.
Bibliography
Van De Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics, 7th ed. Boston: Allyn and Bacon.
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