As an advocate of gifted students within the regular classroom, I work with teachers who may not recognize the attributes of mathematical giftedness, or understand how this ability is manifested. This article describes the general characteristics of mathematically gifted students and how classroom teachers can identify, instruct, and assess students in question through observation, conversation, classroom activities, and individual testing, using true mathematical problems Mathematical problem may mean two slightly different things, both closely related to mathematical games: - general meaning
- a question that can be answered with the help of mathematics ; formal meaning : any tuple (S, C( ), r
geared for the specific student. Cross curriculum indicators of talent are discussed as well as classroom structure and its role in differentiating instruction for mathematically talented students.
Mathematically promising students abound in the public school system, waiting to be recognized, approached, nurtured, and most of all, taught. These students may be called promising, gifted, talented, or academically superior, but the common attribute is that they stand out from their peers and demand to be instructed in the least restrictive environment As part of the U.S. Individuals with Disabilities Education Act, the least restrictive environment is identified as one of the six principles that govern the education of students with disabilities. .
Having the privilege of working with promising young
mathematicians Mathematicians by letter: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z See also
- Requested mathematicians articles
- (by country, etc.)
- List of physicists
External links
provides me a unique opportunity to assess their talent as well as to
nurture NURTURE. The act of taking care of children and educating them: the right to the nurture of children generally belongs to the father till the child shall arrive at the age of fourteen years, and not longer. Till then, he is guardian by nurture. Co. Litt. 38 b. and to advocate for these children. Too often they are swept through classes, and encouraged to skip levels and grades of mathematics instruction in the hope that their distinctive needs will be met, sometimes with no class modifications available. This article provides a general picture of mathematically gifted students and
differentiated instruction Differentiated instruction (sometimes referred to as differentiated learning) is a way of thinking about teaching and learning. It involves teachers using a variety of instructional strategies that address diverse student learning needs. that includes
anecdotal anecdotal /an·ec·do·tal/ (an?ek-do´t'l) based on case histories rather than on controlled clinical trials.
anecdotal adjective Unsubstantiated; occurring as single or isolated event. records of achievement, and also offers curricular suggestions to enhance the teaching and learning of these students within the regular classroom setting.
Identifying the Mathematically
Precocious pre·co·cious
adj.
Showing unusually early development or maturity.
pre·coc
ity , pre·co Identification of mathematically gifted students is not easy. These students may or may not be achieving within the classroom, may or may not show interest, effort, or excitement during math instruction, and may or may not score well on
standardized standardized
pertaining to data that have been submitted to standardization procedures.
standardized morbidity rate
see morbidity rate.
standardized mortality rate
see mortality rate. achievement or
proficiency tests See aptitude tests. . Classroom teachers who have little background in recognizing the attributes of gifted children or who are less knowledgeable in the area of mathematics often mistake hard work for promise. That is to say, teachers who teach the basics in order to provide children with formulas and rules, will not offer the opportunities for mathematically gifted children to demonstrate their distinctive thinking processes that set them apart from good students. Following are some general classroom activities that help us identify exceptional student math abilities.
Attributes of Classroom Activities that Enhance the Identification and Abilities of Mathematically Gifted Students
Mathematically gifted students come in all sizes, ages, and levels of academic achievement. What they have in common, however, is their ability to experience true
problem solvingproblem solving
Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error.
..... Click the link for more information. tasks by internalizing, reshaping, and questioning: multiple strategies to move forward the process of solving the problem. This characteristic is seen if the tasks posed are genuine problems.
What sets these students apart from others is their high threshold of
acknowledgment acknowledgment, in law, formal declaration or admission by a person who executed an instrument (e.g., a will or a deed) that the instrument is his. The acknowledgment is made before a court, a notary public, or any other authorized person. of what constitutes a problem. This may be indicated by
quizzical quiz·zi·cal
adj.
1. Suggesting puzzlement; questioning.
2. Teasing; mocking: "His face wore a somewhat quizzical almost impertinent air" Lawrence Durrell. facial expressionsfacial expression,
n the use of the facial muscles to communicate or to convey mood.
..... Click the link for more information., the inability to see an appropriate initial plan of attack, as well as seeming to not understand exactly what information the problem has given and/or what it is asking. Classroom teachers who suspect they are in the presence of promising mathematicians need to seek out problems that contain an array of discrete levels and can be solved using multiple strategies. If teachers are unsure whether a problem under consideration fits this criteria, they can use the self-test: if they are unsure of how to proceed in order to solve the problem, they are usually in the presence of a true problem. This will be so even with the youngest of children. The teacher, who most likely will be able to solve the problem successfully, will also be able to see the component of multi-layering within the problem that invigorates and sets in motion the creative, mathematical thinking of gifted students.
These children often begin to talk to themselves or to others engaged in the activity, and it is during this processing stage that teachers are
privy One who has a direct, successive relationship to another individual; a coparticipant; one who has an interest in a matter; private.
Privy refers to a person in privity with another—that is, someone involved in a particular transaction that results in a union, to glimpses of mathematical promise. The attributes of the problems described, those that allow for a variety of solutions and thinking paths, provide a type of
intellectual nourishment Noun 1. intellectual nourishment - anything that provides mental stimulus for thinking
food for thought, food
cognitive content, mental object, content - the sum or range of what has been perceived, discovered, or learned that leads students to manifest the great tenaciousness that surrounds the eventual surrender of a solution. With younger students, teacher observation of the process and the surrounding discussion is the most accurate and reliable tool for distinguishing truly gifted students. The students show how they organize knowledge, communicate ideas, practice the art of
conjecture CONJECTURE. Conjectures are ideas or notions founded on probabilities without any demonstration of their truth. Mascardus has defined conjecture: "rationable vestigium latentis veritatis, unde nascitur opinio sapientis;" or a slight degree of credence arising from evidence too weak or too and make convincing arguments.
Providing challenging, thought provoking problems once a week or every other week will
furnish fur·nish
tr.v. fur·nished, fur·nish·ing, fur·nish·es
1. To equip with what is needed, especially to provide furniture for.
2. anecdotal records of student responses to specific problems and multiple strategies that led to solutions. When problems are solved quickly, more challenging ones can be selected until the students are engaged in spirited dialogue concerning the paths to a solution. During these more advanced activities, however, it is important that the mathematical vocabulary is known and understood by the children which may indicate the need for defining or redefining with examples to clear up potential
interpretive in·ter·pre·tive also in·ter·pre·ta·tive
adj.
Relating to or marked by interpretation; explanatory.
in·terpre·tive·ly adv. miscues.
Sample Assessment for the Identification of Young Talented Mathematicians
Jason, a bright fifth grade student from a
neighboring neigh·bor
n.
1. One who lives near or next to another.
2. A person, place, or thing adjacent to or located near another.
3. A fellow human.
4. Used as a form of familiar address.
v. elementary school elementary school: see school. , exhibited mathematical knowledge above the majority of fifth graders. His classroom teacher thought he would be better served as a mathematics student in the sixth grade working with other identified gifted mathematicians.
Jason was transported to my school and became a part of my "mathematically gifted cluster" class. I was surprised and concerned by his lack of vocabulary; he did not know about prime and
composite numbers composite number
n.
An integer exactly divisible by at least one positive integer other than itself or 1.
composite number or factoring, making his integration into the classroom difficult. He was with us for two weeks before I took an hour to administer a problem solving assessment. The problems are taken from Problem solving 2, (1980), Ohio Department of Education. The description of Jason's story includes the
approximation approximation /ap·prox·i·ma·tion/ (ah-prok?si-ma´shun)
1. the act or process of bringing into proximity or apposition.
2. a numerical value of limited accuracy. of grade level appropriateness for intermediate grade students, any history of the problem within my teaching experience, Jason's problem solving strategies, and my commentary.
Problem 1. (Grade 5-6) In the following
subtraction subtraction, fundamental operation of arithmetic; the inverse of addition. If a and b are real numbers (see number), then the number a−b is that number (called the difference) which when added to b (the subtractor) equals problem each digit has been replaced by a given letter. What is the original problem? ABC-CA/AB
This problem was given to the LD inclusion and the noncluster (non-gifted) as a problem of the week. The intent of this problem is to determine a student's understanding of the process of subtraction as well as provide easy access to using variables in an easily substituted problem.
I asked Jason if he had worked this problem before and he answered no. The problem was read to him. He sat quietly for a few minutes appearing to do nothing. I asked him what he was thinking. He said he was thinking what to put in for the numbers. I encouraged him by telling him that he was doing the right thing. He wrote the letter A and said that would equal 2, and B would equal 0, and C would equal 8. I asked him to describe how he could prove he was correct. He looked at me and I prompted that he would substitute the values for the letters into the problem. Without too much time he changed the value of A to 1. I asked him why. He responded that the A in the hundreds place would have to be 1, otherwise there would be a number in the hundreds place in the answer, and there was none. He thought B would still be 0. I suggested he write the problem using just the numbers. When he set the problem up he caught the mistake with C being 8 and promptly changed it to 9. Why? He said that since A was 1, 10 minus 9 would be 1.
Commentary: Although Jason did arrive at the correct answer, he made two critical errors that showed me he has had little opportunity to see the relationships between operations and place value. He also seemed unprepared to use the strategy of substitution to solve an
algebraic equation algebraic equation
Mathematical statement of equality between algebraic expressions. An expression is algebraic if it involves a finite combination of numbers and variables and algebraic operations (addition, subtraction, multiplication, division, raising to a power, and . I suggest this area of patterns be accomplished with heavy emphasis on operations of fractions and decimals as they relate to each other.
Problem 2 (Grade 5-6) One of nine objects, identical in appearance, is slightly heavier than the rest. With only an equal arm balance available, how can the heavier object be determined in just two weighings?
Jason seemed at a loss as to how to go about this problem, which meant this was truly a problem for him. I noticed that he rarely writes down any information in an attempt to solve a problem. I assume he formulates in his head and records things after the fact, which may be why he has to adjust his final answer before it is correct. I
reread Verb 1. reread - read anew; read again; "He re-read her letters to him"
read - interpret something that is written or printed; "read the advertisement"; "Have you read Salman Rushdie?" the question to him and made sure he understood the content of the question and that he knew what a single arm balance was. He said he understood and continued to sit. I suggested that we draw what we knew. I drew 9 circles and a bad sketch of a balance. I asked him if he could figure out how to find the heavier object in just two weighings. He asked if they all looked the same and I said yes. He asked if it couldn't just be picked out from all the rest because it was heavier and I said no, it was only slightly heavier and this could only be picked up by using the balance scale. He pondered about this for quite
awhile a·while
adv.
For a short time.
Usage Note: Awhile, an adverb, is never preceded by a preposition such as for, but the two-word form a while may be preceded by a preposition. . He finally suggested that we could separate the circles into groups of 4, weigh those and if they were equal, you'd know circle number 9 was the heavier. I explained that would work if we were that lucky to divide the items up in that way. Let's assume that does not happen. How could we find out the heavier one with 2 weighings? He then decided to group them into twos. On each pan you'd put two items. If they were equal you would know that the heavier one was not in that grouping. That would be one weighing. He would do the same thing for the other 4. That would be weighing 2. Depending on the results, we still may or may not know which item was heavier, which would mandate a third weighing, which would not help us with this problem. Jason seemed very
frustrated frus·trate
tr.v. frus·trat·ed, frus·trat·ing, frus·trates
1.
a. To prevent from accomplishing a purpose or fulfilling a desire; thwart: and did not seem to have any ideas that would help him solve this problem.
Commentary: This is a good problem for determining reasoning and logic. Jason did seem very confused with even how to go about thinking so as to solve the problem. To use this as a teaching problem I would make sure that a balance scale was available and allow him to test out different ideas. Once the logic is seen, it will stay with him to help him with other problems that are similar. These are the types of problems where a small learning group is the best configuration for the exchange of knowledge. (I have noticed when Jason has been given the opportunity to work with a partner or his table mates, he rarely takes advantage of this. It is possible this could be a cultural issue too).
Problem 3. (Grade 5-6)Place the numbers 1 through 19 into the 19 circles that any three numbers in a row will give the same sum.
I reexplained this to Jason, showing him the constant number and how the rows would add up. He sat and did nothing with his pencil, although I knew he was thinking. I asked him to tell me what he was thinking. He said that he thought the numbers in the outside would go from high to low to balance each other out. I said okay. Even though that is right on thinking he did not make any move to complete the problem. I asked if he wanted a calculator and he said yes. Once the calculator was made available he entered two entries than completed the task correctly. He needed no prompting on this problem. I asked what he entered into the calculator. He said he entered 19 + 2 + 21, then plus the 1 in the middle would equal 22, which he thought would be his target number. He tried one more and then knew he was right and completed the problem.
Commentary: Jason easily saw the relationship between the numbers progressing
clockwise clock·wise
adv. & adj. Abbr. cw.
In the same direction as the rotating hands of a clock.
clockwise
Adverb, adj
in the direction in which the hands of a clock rotate and counter clockwise. I'd like some more explanation from him as to how and why this balance works and are there other number combinations that would work?
Problem 4. (Grade 5-6)A
water lily water lily, common name for some members of the Nymphaeaceae, a family of freshwater perennial herbs found in most parts of the world and often characterized by large shield-shaped leaves and showy, fragrant blossoms of various colors. doubles itself in size each day. From the time the original plant was placed in a pond until the surface was completely covered took 30 days. How long did it take for the pond to be half covered?
Jason immediately picked up the calculator and started doubling the number 2. The number becomes too large for the calculator and Jason shows me that it says Error F. I asked him what he's thinking. He reexplains the problem to me. I asked, if it takes 30 days to cover the entire pond, how many days will it take for half the pond to be covered? He said he didn't have enough information since he didn't know how big the pond was or how big the water lily was. I tried to steer his thinking in a different direction. I reminded him of the first problem we did where letters stood for numbers. I asked him to come up with a letter that would stand for the size of a
pond lily pond lily: see water lily. on day 1. He chose a. I set up a table to show the doubling principle which he already understood. We knew that each day the size doubled so if a= 1, then the chart will look something like this: 1- 1, 2- 2, 3- 4, 4- 8, 4- 16, etc. We related this to using the
exponent exponent, in mathematics, a number, letter, or algebraic expression written above and to the right of another number, letter, or expression called the base. In the expressions x2 and xn, the number 2 and the letter n of 2, 30 times. We knew that the calculator would only take us so far. Before we proceeded I reread the problem one more time. If the pond is covered in 30 days, how many days will it take for half of the pond to be covered? He said it will take 15 days because half of 30 is 15. Although this is a correct statement, its relationship to this problem is often
misunderstood mis·un·der·stood
v.
Past tense and past participle of misunderstand.
adj.
1. Incorrectly understood or interpreted.
2. as having anything to do with the solution. I suggested we continue with our doubling even if we had to do it with paper and pencil. Jason's ability to do addition in his head is superior. Most students would opt to do paper to check, but Jason did not and he did not need to! As we moved to the total of the 29th day and to the 30th day where we both agreed that the pond would be covered, he responded that the pond would be half covered on day 29. He said that because the lily doubled every day, the 29th day would have covered half the pond because it would do its final doubling to cover the pond on day 30.
Commentary: I was
amazed a·maze
v. a·mazed, a·maz·ing, a·maz·es
v.tr.
1. To affect with great wonder; astonish. See Synonyms at surprise.
2. Obsolete To bewilder; perplex.
v.intr. with the speed and accuracy of Jason's mental addition. This is not one of the skills we value as much as other countries. What I did notice was that although Jason could do arithmetic, he seemed unaware of the bigger picture. That is to say, what and why am I doing this? This was not an addition problem. It was a reasoning problem that dealt with the relationship of a doubling number and size. There were many shorter strategies he could have taken and arrived at the same idea if he had seen the relationships. It is my
hunch hunch
n.
1. An intuitive feeling or a premonition: had a hunch that he would lose.
2. A hump.
3. A lump or chunk: "She . . . that Jason has not has these types of experiences and desperately needs them.
Conclusion Commentary
Based upon the hour long assessment of grade appropriate problems, it is my opinion that Jason is bright and works easily with basic math facts. He does seem to be
deficient de·fi·cient
adj.
1. Lacking an essential quality or element.
2. Inadequate in amount or degree; insufficient.
deficient
a state of being in deficit. in using appropriate mathematical vocabulary in that not having immediate recall of terms such as
numerator numerator
the upper part of a fraction.
numerator relationship
see additive genetic relationship.
numerator Epidemiology The upper part of a fraction and
denominator denominator
the bottom line of a fraction; the base population on which population rates such as birth and death rates are calculated.
denominator makes analyzing relationships and patterns difficult. It is also my conclusion that Jason has not been exposed to real problems nor has he been made to explain his thinking using written or oral communication.
I suggest that Jason be given problems that involve the understanding of relationships as an introductory teaching assessment. He needs to explain how and why he arrived at solutions. Based upon his response, the teacher may opt for reteaching from a text, more problems that use a similar strategy and cover the same content, or higher level problems that really call for Jason to use a variety of knowledge. He does need instruction. I might suggest keeping him with the class some of the time and giving him challenge worksheets that go with our text. I might also suggest that he be assigned the sections of the text that deal with understanding, not the technical aspects of arithmetic.
Cross-Curriculum Indicators of Mathematical Talent
Other reliable indicators of mathematically gifted students occur across the curriculum. Talented mathematicians see the world in mathematical ways. The propensity to
assimilate as·sim·i·late
v.
1. To consume and incorporate nutrients into the body after digestion.
2. To transform food into living tissue by the process of anabolism. and
synthesize To create a whole or complete unit from parts or components. See synthesis. all content via mathematical language can be used as a benchmark that identifies students as mathematically talented.
Being
attuned at·tune
tr.v. at·tuned, at·tun·ing, at·tunes
1. To bring into a harmonious or responsive relationship: an industry that is not attuned to market demands.
2. to the use of logic and sequencing in their daily and/or creative writing samples, sensing the necessity of precision when setting up tables for data or organizing information, is a reliable reference point that certain students must process and interpret their world through the lens of mathematics.
Social Studies provides a rich arena to observe the various types of thinking exhibited by students. The following activity offers the mathematically gifted ample opportunity to produce a work product using their unique thinking abilities.
Problem: In social studies, we studied the effects of two-children families versus three children families in population education. Your task is to create a model of two-child family growth versus three-child family growth. Assume that two couples, age 20, without children, are placed upon the island of Fertilia. These couples and all of their
descendants DESCENDANTS. Those who have issued from an individual, and include his children, grandchildren, and their children to the remotest degree. Ambl. 327 2 Bro. C. C. 30; Id. 230 3 Bro. C. C. 367; 1 Rop. Leg. 115; 2 Bouv. n. 1956.
2. are committed to having only two children per family. At the same time, two other couples, also age 20 and without children, are placed upon the neighboring island of Greater Fertilia. They and their descendants agree to have exactly three children per family. Show how the population will grow on these two islands over a period of 160 years. Make whatever assumptions you wish about when children are born, when people die, and so forth. But remember, that people die as well as being born and, therefore, have to be accounted for.
Students engaged in this assessment activity are asked to use mathematical skills as well as predictive logic. The term "model" evokes a variety of possible outcome products, depending upon the student's areas of multiple intelligence, as described by
Howard Gardner Howard Gardner, born on July 11, 1943 in Scranton, Pennsylvania, is a psychologist who is based at Harvard Graduate School of Education. He is best known for his theory of multiple intelligences[0]. In 1981, he was awarded a MacArthur Prize Fellowship. .
Teachers must be
astute as·tute
adj.
Having or showing shrewdness and discernment, especially with respect to one's own concerns. See Synonyms at shrewd.
[Latin ast observers of their students. Students who choose to represent information with graphs, for example, envision the world in a linear, mathematical way and use quantitative information to support their reasoning. Teachers also need to determine if children are able to take information given to them, or self selected information and transform it into a variety of appropriate numerations. Do they usually create a variety of graphs to show a solution? Can they explain processes using manipulatives as well as the more abstract paper and pencil form? Do they question the teacher by asking "what if" and "why" questions, and are they willing to do some investigation to prove their point and share it with others? When teachers answer yes to the majority of these questions, then not only are they most likely in the presence of mathematically promising students, but the teacher has taken a large step toward understanding the necessity for differentiated instruction for these students, and successfully enhancing their instruction of gifted students, as well as other students, in the class.
Discussion
Working with promising math students can be a
daunting daunt
tr.v. daunt·ed, daunt·ing, daunts
To abate the courage of; discourage. See Synonyms at dismay.
[Middle English daunten, from Old French danter, from Latin task. The general education curriculum does not speak to the needs of this group, although there are sections in most text series that delineate
enrichment enrichment Food industry The addition of vitamins or minerals to a food–eg, wheat, which may have been lost during processing. See White flour; Cf Whole grains. activities, or identify material as "for more challenge." This material is most often appropriate for the hard working student, but not for the mathematically gifted student because the activities are generally too one-dimensional. Thus, the task of finding appropriate material falls to the teacher.
One way to learn about proven instructional techniques is through professional journals published by the
National Council of Teachers of Mathematics The National Council of Teachers of Mathematics (NCTM) was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally. . Using a curriculum supervisor or
gifted education Gifted education is a broad term for special practices, procedures and theories used in the education of children who have been identified as gifted or talented. Programs providing such education are sometimes called Gifted and Talented Education (GATE) or teachers if available in one's district is also an excellent place to begin the search for academically challenging activities. Materials
germane ger·mane
adj.
Being both pertinent and fitting. See Synonyms at relevant.
[Middle English germain, having the same parents, closely connected; see german2. to differentiated instruction for mathematically gifted students may also be discovered by searching outside a grade level area such as math books specifically targeted for levels two to three grade levels above the norm. The State Department of Education may also have monographs that target problem solving and the teaching of specific strategies. And sometimes the best resources are within your own school, utilizing colleagues' ideas that have worked with older students. More importantly, true gifted students should not be given a math text and allowed to progress at their own pace. Like all students, these students require care, nurturing, and teaching.
Clustering like-minded students together for math allows peer interaction as well as a small group for instruction during regular math class. Students could be given a problem to cooperatively solve while other math instruction in occurring; however, in this case it is essential that time be made available for discussion and the connection of the problem to the appropriate mathematical concept under investigation. Maintaining journals that capture successes and miscues during the mathematical journey provides assessment records as well as important information that should be shared with future math teachers. For younger children or learning disabled children with exceptional math ability, tape recording discussions and explanations is also a viable method of assessment. It is not so much a function that a correct answer is given, but the richness of the dialogue and connections made throughout the working process that will tell the progress made by the individual student.
Working with mathematically promising students is a tiring and difficult, but
joyous joy·ous
adj.
Feeling or causing joy; joyful. See Synonyms at glad1.
joyous·ly adv. experience. Teachers must first provide challenging activities that capture the interest and mind of the student, then observe, listen, and identify the variety of strategies and depth of knowledge base used in an attempt to solve problems. A continuous diet of open-ended or nonroutine problems will yield the most reliable information concerning mathematically promising children. Once children are identified, the search for materials, research, and on site help become the focus of the teacher. Finally, using techniques such as clustering as well as fostering cooperative group activities will encourage and strengthen not only the students' mathematical knowledge, but the teacher's appreciation of true mathematical gifts. We stand in awe of mathematical genius and the promise we are privileged to
behold be·hold
v. be·held , be·hold·ing, be·holds
v.tr.
1.
a. To perceive by the visual faculty; see: beheld a tiny figure in the distance.
b. .
Marilyn Hoeflinger is a 25-year veteran of public school teaching. She teaches math and science to sixth grade students in The Hilliard City School District, Ohio and is a doctoral candidate at The
Ohio State University Ohio State University, main campus at Columbus; land-grant and state supported; coeducational; chartered 1870, opened 1873 as Ohio Agricultural and Mechanical College, renamed 1878. There are also campuses at Lima, Mansfield, Marion, and Newark. .
Manuscript submitted July, 1997. Revision accepted February, 1998.
Comre from:http://www.thefreelibrary.com/Mathematics+and+science+in+gifted+education:+Developing...-a021225361
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