During my practicum experience I have had the chance to teach a lesson on adding/subtracting decimals at the sixth grade level. We used a checkbook register to replicate a real life experience and gave students practical knowledge.
The objectives outlined for the class were:
1) Students will be able to add and subtract decimals and justify their steps.
2) Students will be able to use a checkbook register to add deposits and subtract purchases from their previous balance.
The strategy I used was an "I do, you do." Because of time there was no real "we do." I modeled how to use the register to add deposits to previous balances and how to subtract purchases, but then students were to get together with their elbow partner to add and subtract additional deposits and purchases.
For students that understood the concept really well, they were able to add and subtract without much further guidance, but those who really did not understand what a checkbook registrar was and what it was used for it was difficult. The two ELL students in the class really struggled with the concepts of money as decimals. I do not think that they use decimals to represent their money, so it was difficult for them to understand that is what we did. If I were to change the lesson, I would go back and give more of a history and what a registrar was used for. I think giving background knowledge would help with the whole process and lesson. I would also talk about how our money is broken up and that it is represented by decimals. I did not even think of this being an issue when I started the lesson. Next time I will be more aware.
To check for understanding I collected the registrars and looked through each of them to make sure the addition and subtraction was done accurately (this was possible because the class only had 12 students). I was able to see some common mistakes like .40 + .70 = .11 instead of 1.10. The next day I addressed some of the math fallacies and hopefully as we continue with adding and subtracting fractions students will overcome the fallacies and understand the logic properly.
Sunday, October 30, 2011
Sunday, October 16, 2011
Warm-ups in Math Education
Warm-ups in math I think are imperative and should be done everyday which will promote a routine in the classroom as well as helps focus students when walking into class. We as teachers middle and high school teachers have less control of what goes on with our students before walking in our door; we typically get them one period of the day as opposed to being a self-contained classroom. Warm-ups can bring routine which can potentially calm the students down and get them focused on the math lesson ahead of them.
I think warm-ups should be short, not spending the whole class period doing them. They should also be something relatively simple, like a couple questions to review previous lessons. I think if we use warm-ups as a sort of pre-test or give them something beyond what they have learned it might deter them from trying or engaging in the lesson that follows.
Also, I think warm-ups should be an individual activity so that students can personally assess where they are in learning and understanding the material.
Additionally, I will make sure warm-ups are gone over and that we do not just give the answers to the question but will give the answer and demonstrate multiple ways if possible to see how one can get to that answer. It is valuable that students see that there is no one direct path to the right answer and that it is okay to get to the answer in multiple ways.
I think warm-ups are very important and can provide to be a great tool!
I think warm-ups should be short, not spending the whole class period doing them. They should also be something relatively simple, like a couple questions to review previous lessons. I think if we use warm-ups as a sort of pre-test or give them something beyond what they have learned it might deter them from trying or engaging in the lesson that follows.
Also, I think warm-ups should be an individual activity so that students can personally assess where they are in learning and understanding the material.
Additionally, I will make sure warm-ups are gone over and that we do not just give the answers to the question but will give the answer and demonstrate multiple ways if possible to see how one can get to that answer. It is valuable that students see that there is no one direct path to the right answer and that it is okay to get to the answer in multiple ways.
I think warm-ups are very important and can provide to be a great tool!
Sunday, October 2, 2011
Standards, Standards Everywhere
I personally looked at the standards for numbers and operations.
Here are NCTM's:
Grades 6–8 Expectations: In grades 6–8 all students should–
work flexibly with fractions, decimals, and percents to solve problems;
compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
develop meaning for percents greater than 100 and less than 1;
understand and use ratios and proportions to represent quantitative relationships;
develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation;
use factors, multiples, prime factorization, and relatively prime numbers to solve problems;
develop meaning for integers and represent and compare quantities with them.
Common Core:
6th Grade: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
Compute fluently with multi-digit numbers and find common factors and multiples.
Apply and extend previous understandings of numbers to the system of rational numbers.
7th Grade: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
8th Grade: Know that there are numbers that are not rational, and approximate them by rational numbers.
Salem-Keizer's:
umber and Operation Goals
Number Sense
Use numbers in various forms to solve problems (6, 7, 8)
Understand and use large numbers, including in exponential and scientific notation (6, 7, 8)
Reason proportionally in a variety of contexts using geometric and numerical reasoning, including scaling and solving proportions (6, 7, 8)
Compare numbers in a variety of ways, including differences, rates, ratios, and percents and choose when each comparison is appropriate (6, 7, 8)
Order positive and/or negative rational numbers (6, 7, 8)
Express rational numbers in equivalent forms (6)
Make estimates and use benchmarks (6, 7, 8)
Operations and Algorithms
Develop understanding and skill with all four arithmetic operations on fractions and decimals (6)
Develop understanding and skill in solving a variety of percent problems (6)
Use the order of operations to write, evaluate, and simplify numerical expressions (7, 8)
Develop fluency with paper and pencil computation, calculator use, mental calculation, estimation; and choose among these when solving problems (6, 7)
Properties
Understand the multiplicative structure of numbers, including the concepts of prime and composite numbers, evens, odds, and prime factorizations (6)
Use the commutative and distributive properties to write equivalent numerical expressions (7, 8)
____________
When looking at the standards it looked like NCTM's were the shortest and most vague of the standards. Both the Salem-Keizer district and the CC were pretty specific and went into specifically which grade level different standards were expected in. NCTM simply breaks down the standards for middle school grades (6-8) as whole. I thinking having to refer back to the NCTM, CC and school district standards will have a tendency to cause some confusion. Although there are definitely veins of similarity between the standards, I think I would have a tendency to look at the school district and CC as opposed to NCTM when designing lesson plans. I think that when you look at NCTM and because it only gives an overarching standard for middle school it will be a little more difficult to break down according to grade. CC and at least the Salem-Keizer district have standards for the each grade level which I personally think will make lesson planning a tad bit easier.
Here are NCTM's:
Grades 6–8 Expectations: In grades 6–8 all students should–
work flexibly with fractions, decimals, and percents to solve problems;
compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
develop meaning for percents greater than 100 and less than 1;
understand and use ratios and proportions to represent quantitative relationships;
develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation;
use factors, multiples, prime factorization, and relatively prime numbers to solve problems;
develop meaning for integers and represent and compare quantities with them.
Common Core:
6th Grade: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
Compute fluently with multi-digit numbers and find common factors and multiples.
Apply and extend previous understandings of numbers to the system of rational numbers.
7th Grade: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
8th Grade: Know that there are numbers that are not rational, and approximate them by rational numbers.
Salem-Keizer's:
umber and Operation Goals
Number Sense
Use numbers in various forms to solve problems (6, 7, 8)
Understand and use large numbers, including in exponential and scientific notation (6, 7, 8)
Reason proportionally in a variety of contexts using geometric and numerical reasoning, including scaling and solving proportions (6, 7, 8)
Compare numbers in a variety of ways, including differences, rates, ratios, and percents and choose when each comparison is appropriate (6, 7, 8)
Order positive and/or negative rational numbers (6, 7, 8)
Express rational numbers in equivalent forms (6)
Make estimates and use benchmarks (6, 7, 8)
Operations and Algorithms
Develop understanding and skill with all four arithmetic operations on fractions and decimals (6)
Develop understanding and skill in solving a variety of percent problems (6)
Use the order of operations to write, evaluate, and simplify numerical expressions (7, 8)
Develop fluency with paper and pencil computation, calculator use, mental calculation, estimation; and choose among these when solving problems (6, 7)
Properties
Understand the multiplicative structure of numbers, including the concepts of prime and composite numbers, evens, odds, and prime factorizations (6)
Use the commutative and distributive properties to write equivalent numerical expressions (7, 8)
____________
When looking at the standards it looked like NCTM's were the shortest and most vague of the standards. Both the Salem-Keizer district and the CC were pretty specific and went into specifically which grade level different standards were expected in. NCTM simply breaks down the standards for middle school grades (6-8) as whole. I thinking having to refer back to the NCTM, CC and school district standards will have a tendency to cause some confusion. Although there are definitely veins of similarity between the standards, I think I would have a tendency to look at the school district and CC as opposed to NCTM when designing lesson plans. I think that when you look at NCTM and because it only gives an overarching standard for middle school it will be a little more difficult to break down according to grade. CC and at least the Salem-Keizer district have standards for the each grade level which I personally think will make lesson planning a tad bit easier.
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