Keywords: Activities, ,
Ref: Chris1
Author(s): Nkagomi, Koji
Date: 2000
Title: Gathering Circles: An Experience in Creativity and Variety
Journal or Publisher: Mathematic teacher journal
Volume, Issue, Pages: Vol 93 Num 9, p. 746-752
Reviewer: Chris
Date of Review: 2-13-01
The teacher in this article was concerned with a student in her ninth grade class in Japan whom was doodling with a compass instead of paying attention to her lesson. She was also concerned with the surprising high percentage of Japanese students whom reported a dislike of math. Paying attention to the student’s doodling, she noticed that it repeated an interesting mathematical pattern. Starting from a set of tiles in a triangular shape it spread outward in a hexagonal pattern. The teacher decided to integrate the student’s pattern into the lesson plan in order to maintain interest, encourage creative thought, look at the problem from many angles and to express the logic of their thoughts. She used what she called an "open-ended approach."
In the open-ended approach an incomplete problem was presented at first. The students began by brainstorming relationships in the pattern. On the first day, the students were to think of ideas and develop problems. They questioned and made discoveries. On the second day, the students further developed their ideas so that they could express their thoughts to others through mathematical rules and proofs to explain their discoveries. On the third day small groups presented their ideas to the class and answered questions. The article presents many of the ideas which the students came up with and some were very impressive, especially for ninth graders.
At first, I wondered where the teacher was going with this or how it would fit in with a particular unit or where she was going with this. Was there any clear material objective? However, I then thought that this was a really good idea, especially when looking at some of student presentations. I do think it would take a lot of teacher prompting and would find it hard to grade both for the discussed creative or artistic element but also because the objectives or what the teacher wanted might be hard for students to come to terms with.
Keywords: Technology, ,
Ref: Chris2
Author(s): Williams, Susan; Copley, Juanit
Date: 1994
Title: Promoting Classroom Dialogue: Using Calculators to Discover Patterns in Dividing Decimals
Journal or Publisher: Mathmatics Teaching in the Middle School
Volume, Issue, Pages: Vol 1, Num 1, April 1994
Reviewer: Chris
Date of Review: 2/20/01
In this article, the teacher struggled with when to use calculators in the classroom. Must a student understand a concept before using a calculator? She had read that calculators had greater power as a teaching tool than as merely a checking tool. But she did not know how to teach a concept with a calculator.
Relied on assistance from university researchers, she presented her students with a list of four division problems in
which the divisor remained constant but the dividend varied in size by movement of the decimal place. They were
asked to answer (1) what was the same, (2)what differed, (3)what patterns existed, and (4) what caused these
patterns to occur. Students were to verify their predictions with long division and use calculators to check this
with their calculators
She slowly got some responses after initial blank stares. Her main concerns were that they did not make it to estimating quotients, and she did not think that she used the calculator well.
For the second class the opening question was "If you know 2726/58= 47, what would you guess 22.6/58 is equal to? Students were then allowed to investigate several examples and use calculators to assist in the computation. They were to answer the same four questions as the first class. They were more excited then the first class in the patterns that they had discovered and the rules they had developed.
She then moved to ask estimations asking how much 6174/98 is. Given its approximation of 60 by rounding up 98, they were asked to predict an approximation of 617.4/98. Letting the groups respond to here what if questions on estimation problems, the students ended up creating their own problems which deviated from the original plan but was exciting to the teacher. Then they asked what happens if you move the decimal place in both dividend and divisor? A student then interjected an explanation of "WHY we move the decimal points" by showing the problems as he had written them.
The teachers discovery was that students’ own investigations and search for patterns led them into richer mathematics territory. The teacher felt that when students the students were encouraged to experiment and to ask questions, they constructed a deeper understanding. In addition, she found that the calculator could provide computational power often needed to discover patterns when working with interesting numbers.
This article seems very familiar to the last in that it promoted an open exploration. Students developed their own ideas and these iidas were again very good.
Keywords: Algebra, Connections,
Ref: Chris3
Author(s): Woodbury, Sonia
Date: 2000
Title: Teaching Towards the Big
Ideas of Algebra
Journal or Publisher: Mathematics
Teaching in the Middle School
Volume, Issue, Pages: Vol 6, No 4,
226-231
Reviewer: Chris
Date of Review: 2/01
This article looks at a need for students to gain both procedural knowledge and broadly connected conceptual knowledge. It moves from an instrumental understanding of algebra that has held rules without reasons to a relational understanding that consists of building up conceptual structures. The article looks to approach mathematics as a whole with students acquiring a sense of big ideas and how they are interrelated. It contrasts teaching Math topics in the table of contents, but instead promotes combining procedural with conceptual ideas to broaden the students understanding and range of applicability.
For an example, this article looks at students understanding of prime numbers stating that students normally come to identify prime numbers by the definition that they are only divisible by one and themselves. They are then taught to decompose a prime number into its prime factors, usually by making trees. But the author feels that this is not connected with a broader understanding of primeness or of when the idea might be useful for tasks other than finding common denominators for fractions.
The goals of the teachers method was to learn about reasoning abilities and discuss prime numbers in a larger conceptual arena that included ideas about numbers and number systems.
It began with a story problem of how to arrange 120 plants in garden 1 so that there was the same number of plants in each row. It was then asked how they knew that they had all the possibilities. Then, the students were then given 135 plants to arrange in garden 2 and were to explain the possible ways of arranging plants in both gardens so that there were the same number of rows. The students developed strategies, some recited rules (such as only even numbers are divisible by two) etc.
Extension problems were then given asking what are prime numbers, what are the prime numbers that divide both 120 and 135 and what is the largest number that divides both 120 and 135. They were then asked to find a relationship between the greatest common divisor and the prime factors. Students discovered that the common factors included the primes and their product 15.
They were given a similar plant problem. A student example did not use terms prime factor and common factor but referred to the original problem of finding rows for gardens. The article discussed the possibility of extending ideas through different problem contexts, but in their strategy they extended the idea of primeness by another problem in which the students were to look at ways of making 5 through fractions. They are asked why they sometimes think of 5 as a prime factor and otherwise as having many factors.
The students were interested in something they had previously only known as a rule. At first, one student commented unless in the chapter in your text on fractions 5 is usually a prime number. She relied on separate text book chapters to differentiate number systems. Then, another student explained that you can always divide fractions in half.
The students came to demonstrate relational understanding and discuss connections in meaningful ways. In addition, by discussing the specific topic (prime numbers) in a larger conceptual arena of ideas about numbers and number systems, it was thought that students could build ideas in ways not considered before.
I would agree that there is a lot lost in
merely teaching rules and procedures and
I would agree with the ideas of the
article. I also find it interesting how
students might arrange material. It would
take inovation to arange the class so
that this type of learning was enforced
and rewarded.
Keywords: Teaching Strategies
Ref: Chris4
Author(s): Henningsen, Marjorie
Date: 2000
Title: Triumph through Adversity:
Supporting High-level Thinking
Journal or Publisher: Mathmatics
Teaching in the Middle School
Volume, Issue, Pages: Vol 6, No 4, Dec
2000
Reviewer: Chris
Date of Review: 2/25/01
This article looks at the difficulty of enacting challenging, or higher level tasks in the classroom. It looks at tasks that engage students in higher level cognitive functions (looking for patterns, explaining ideas, justifying conclusions, testing conjectures, framing and posing problems, and making decisions) and how they bring students to learn more than their counterparts where such tasks are not performed. It also looks at how such higher level tasks usually do not meet the teachers original intentions. Yet, it discusses how struggling with high-level thinking and reasoning can benefit students even if they are not completely successful in solving the tasks. It also looks at how a teacher can reflect on seemingly unproductive task in one lesson and look for ways to improve instruction on subsequent days.
In an example exercise, students were to recognize and generalize patterns for odd and even figures and write formulas to describe them. The students did not make significant progress towards accomplishing the goal of the task. The students made observation about difference, but were unable to articulate the patterns clearly enough to generate verbal or symbolic formulas to describe the patterns.
The article lists various factors that both support and inhibit higher-level thinking, reasoning and sense making. For this particular case, the task may have been inappropriate: the students may lack prerequisite understanding or many students, and many could not articulate the patterns verbally or symbolically without the teachers help. The teacher also did not consistently hold students accountable for high-level products or processes: the teacher did not press the students to give high quality answers that reflected thoughtful reasoning and asked questions before students had time to arrive at deeper answers.
On the second day after reflection, looking at logs, the teacher realized that he had not held students accountable for explaining their thinking. He also realized that some groups had observed patterns that might help other students move on to verbal and symbolic generalizations. He therefore began the day by having students report their observations and then had the students work in groups again.
By the end of class most groups had determined the pattern and were able to find the volume of any figure given the figure number. Some groups were on their way to writing formulas, some with difficulty, but most closing in on correct formulas.
The article lists things that helped students engage in higher level thinking on the second day. The teacher consistently questioned the students and pressed them to explain their reasoning. Rather than rushing through his questions and student responses, he used questioning techniques that were more productive in eliciting better responses. He encouraged students to test their ideas and refine their articulation of the patterns. He also helped students complete tasks without removing complexity. This scaffeling was done through the kinds of questions he asked and in his allowance of students to share their observations which brought more resources without fewer demands of the task. Students also had sufficient time to grapple with the problematic aspects of the task. They were allowed to work through their confusion on their own and find a successful solution.
This article was good both for its look at student learning and for its lessons to beginning teachers. Reflective teaching and struggling through adversity in the development of deeper mathematical or teaching understandings is educational, reassuring, and inspiring. They do not have to and it might at times it may be better if they do not seem to get it right away. On a negative note, I was drawn to the words MOST students were able to in the article. The list of Factors that support or inhibit higher level thinking could also be a good thing to have around.