Since the beginning of this school year, I have wanted to collaborate with teachers in my building for some type of problem solving "club" where we can work together on a mathematical task. My original plan was to have this during lunch once or twice per month and invite teachers to discuss a problem we could work on throughout the month. As so often happens in our school lives, schedules interfered and I realized it would be difficult to schedule these meetings during lunch. I tried to think of a way to involve as many teachers in our building as possible without requiring additional meetings.
As I've been engaging in some productive struggle of my own about this situation, I have also been working to plan the elementary school methods for mathematics course that I begin teaching this week. I thought how wonderful it would be if the students in my class could discuss some of these rich mathematical tasks with others outside of our small classroom.
Finally - all of these thoughts merged together! Many of the teachers in our building participated in the Global Read Aloud that was started by Pernille Ripp. Wouldn't it be amazing to do something similar - but this time in mathematics?!?
I am inviting you to participate in our Global Engagement Task in Math (GETMath, for short). For the month of February, I have selected a problem from Marilyn Burns about sums of consecutive numbers. The object of the task is to find patterns when adding consecutive numbers. Think about what patterns you find, what surprises you, and anything else you notice!
I encourage you to begin the problem on your own - then talk with others about it! Share your thinking and your findings. Compare your results with what others have found. Engage in some productive struggle!
You can also share your thoughts on Twitter at #GETMath. I hope you will join me in this experiment! You can learn more here: http://bit.ly/1eybFKx
Wednesday, January 29, 2014
Tuesday, January 21, 2014
Productive Struggle
When working with students in math, it is important to engage them in productive struggle. Through productive struggle, students feel that learning goals are attainable and the effort necessary to attain these learning goals is worthwhile. It creates a sense of hope and students feel empowered with an increased sense of efficacy.
I love this quote about productive struggle: "Basically, academic rigor is helping kids learn to think for themselves."
If we use the above quote to frame our thinking about how students might approach a math problem, it is important that the teacher not make assumptions about the student's understanding. Using questions to prompt student thinking is a powerful way to have students use metacognition - thinking about their own thinking.
Some of the questions I frequently use include:
I love this quote about productive struggle: "Basically, academic rigor is helping kids learn to think for themselves."
If we use the above quote to frame our thinking about how students might approach a math problem, it is important that the teacher not make assumptions about the student's understanding. Using questions to prompt student thinking is a powerful way to have students use metacognition - thinking about their own thinking.
Some of the questions I frequently use include:
- How might you begin?
- What do you already know?
- What do you need to do next?
- Why did that happen?
- What have you tried? What happened?
- Show me how this will work on the next problem.
- Why did you ______?
- What do you think will happen?
- What might you try?
- How did you decide this answer is correct?
Answering a student's question with a question of your own allows you - the teacher - to help the student explain his/her thinking and develop his/her understanding of mathematics.
Monday, January 13, 2014
Mastery of Mathematics - What does it look like?
Recently, I read Grant Wiggins' article "Getting Students to Mastery: How Good is Good Enough?" The title alone for this article as made me think about several conversations with teachers over the past few weeks. Often, in mathematics, we refer to mastery at the elementary school level as quick recall of basic facts. But mathematics is so much more.....
While Common Core and earlier sets of standards have given educators an idea of "what" needs to be taught, the 2001 National Research Council report Adding It Up provides a clear description for mathematical proficiency. In their definition, mathematical proficiency is comprised of five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.
As shown in the picture above, the five strands are intertwined. No particular strand takes precedence over another strand - and the strands are interdependent. No single strand can exist in isolation. In their description of mathematical proficiency, Kilpatrick, Swafford, and Findell write, "How learners represent and connect pieces of knowledge is a key factor in whether they will understand it deeply and can use it in problem solving" (p. 117) and "Learning with understanding is more powerful than simply memorizing because the organization improves retention, promotes fluency, and facilitates learning related material" (p. 118).
This brings me back to the article by Grant Wiggins. In the article, Wiggins proposes the following definition for mastery:
Mastery is effective transfer of learning in authentic and worthy performance. Students have mastered a subject when they are fluent, even creative, in using their knowledge, skills, and understanding in key performance challenges and contexts at the heart of that subject, as measured against valid and high standards.
The Common Core State Standards for Mathematics provide the standards for us to use as the measurement. Our job as educators is to provide a balance of learning opportunities that will engage students and require them to use all five strands of mathematical proficiency in their learning. Today's learners need authentic learning tasks to master the mathematical content. Imagine how many people might say "Math was my favorite subject in school!" if we had received different instruction when we were students!
While Common Core and earlier sets of standards have given educators an idea of "what" needs to be taught, the 2001 National Research Council report Adding It Up provides a clear description for mathematical proficiency. In their definition, mathematical proficiency is comprised of five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.
As shown in the picture above, the five strands are intertwined. No particular strand takes precedence over another strand - and the strands are interdependent. No single strand can exist in isolation. In their description of mathematical proficiency, Kilpatrick, Swafford, and Findell write, "How learners represent and connect pieces of knowledge is a key factor in whether they will understand it deeply and can use it in problem solving" (p. 117) and "Learning with understanding is more powerful than simply memorizing because the organization improves retention, promotes fluency, and facilitates learning related material" (p. 118).
This brings me back to the article by Grant Wiggins. In the article, Wiggins proposes the following definition for mastery:
Mastery is effective transfer of learning in authentic and worthy performance. Students have mastered a subject when they are fluent, even creative, in using their knowledge, skills, and understanding in key performance challenges and contexts at the heart of that subject, as measured against valid and high standards.
The Common Core State Standards for Mathematics provide the standards for us to use as the measurement. Our job as educators is to provide a balance of learning opportunities that will engage students and require them to use all five strands of mathematical proficiency in their learning. Today's learners need authentic learning tasks to master the mathematical content. Imagine how many people might say "Math was my favorite subject in school!" if we had received different instruction when we were students!
Tuesday, January 7, 2014
Building Knowledge about Fractions
Fractions. Did that make your stomach hurt? :-)
Many people often view fractions as being completely separate from whole numbers. Within Common Core, students should be making connections with what remains true about whole numbers and what might be unique to fractions. For example, the number 4 could also be represented as 16 fourths or 4/1. Recognizing these representations as being equivalent is an important understanding for students.
In the 2010 issue of NCTM's Teaching Children Mathematics, Wendy S. Bray and Laura Abreu-Sanchez describe in their article "Using Number Sense to Compare Fractions" how they have introduced fractions to third-grade students. They recommend using circles when introducing fractions. Their reasoning for using circles rather than other area models is that the missing piece is evident.
This was a BIG a-ha moment for me when I read this article. For many years, I have introduced fractions by having students build a fraction kit based on lessons from Marilyn Burns. What I had not considered was that the students would need to know the size of the whole in order to think about the relative size of the fractions we built. Because the lesson always started from the whole (a strip of paper) and we then began to divide the strips into pieces, it was evident to most students that 1/2 was represented by half of a strip of paper. However, this was only because the students had seen the initial strip of paper. I love that using a circle allows students to use intuition and think more deeply about fraction sense.
From using the circle, the students branched out to other representations with fractions - such as area models, number lines, and linear models. By continuing to use these representations to explore different ideas about fractions - such as determining whether a fraction was more or less than one-half - students were able to connect their understanding with the symbols used to represent the fractions.
More about fractions coming later..... In the meantime, this is a great article to get you started with teaching fractions!
Many people often view fractions as being completely separate from whole numbers. Within Common Core, students should be making connections with what remains true about whole numbers and what might be unique to fractions. For example, the number 4 could also be represented as 16 fourths or 4/1. Recognizing these representations as being equivalent is an important understanding for students.
In the 2010 issue of NCTM's Teaching Children Mathematics, Wendy S. Bray and Laura Abreu-Sanchez describe in their article "Using Number Sense to Compare Fractions" how they have introduced fractions to third-grade students. They recommend using circles when introducing fractions. Their reasoning for using circles rather than other area models is that the missing piece is evident.
This was a BIG a-ha moment for me when I read this article. For many years, I have introduced fractions by having students build a fraction kit based on lessons from Marilyn Burns. What I had not considered was that the students would need to know the size of the whole in order to think about the relative size of the fractions we built. Because the lesson always started from the whole (a strip of paper) and we then began to divide the strips into pieces, it was evident to most students that 1/2 was represented by half of a strip of paper. However, this was only because the students had seen the initial strip of paper. I love that using a circle allows students to use intuition and think more deeply about fraction sense.
From using the circle, the students branched out to other representations with fractions - such as area models, number lines, and linear models. By continuing to use these representations to explore different ideas about fractions - such as determining whether a fraction was more or less than one-half - students were able to connect their understanding with the symbols used to represent the fractions.
More about fractions coming later..... In the meantime, this is a great article to get you started with teaching fractions!
A Sunshine Award....
This past weekend, I was nominated for a Sunshine Award by my principal Jacki Prati - which was a pleasant distraction from the cold! Jacki is a exceptional principal and truly leads by example! You should definitely check out her blog: http://teacheratheart.weebly.com/ Before becoming a principal, Jacki was one of the literacy coaches in our district and - while I certainly love math - I always appreciate the books and love of literacy she brings to any conversation!
Cathy Mere - another literacy guru in our district - describes the rules for the Sunshine Award on her blog (http://reflectandrefine.blogspot.com/):
The description of the Sunshine Award is shared by Matt Renwick:
The Sunshine Award gives others an opportunity to learn more about me as a blogger and then, in turn, I will send sunshine the way of 11 other amazing bloggers for you to get to know!
and the rules:
- Acknowledge the nominating blogger.
- Share 11 random facts about yourself.
- Answer the 11 questions the nominating blogger has created for you.
- List 11 bloggers. They should be bloggers you believe deserve some recognition and a little blogging love!
- Post 11 questions for the bloggers you nominate to answer and let all of the bloggers know they have been nominated. (You cannot nominate the blogger who nominated you.)
My first thought was that this will be MUCH more than I usually post - but I love the idea of sharing with other bloggers. So here it is....
11 Random Facts about Me:
1. Math was NOT my first love. I read voraciously as a child - and I still love to curl up with a good book. However, I enjoy teaching math more than I enjoy teaching language arts.
2. In college, I was the director of advertising for our student newspaper. Basically, I worked my way up the ladder after answering an ad to sit in the office to answer the phone. It was an easy way to earn a little spending money - and made me appreciate how much effort goes into creating a newspaper. I proofed the ads for our paper while several aspiring journalists were frantically trying to meet deadlines.
3. My husband and I were in the same 5th grade class. I like to remind students to be kind to everyone because you never know who you'll wind up with in life. We didn't talk in 5th grade but started dating in high school.
4. One of the things I noticed immediately about the Sunshine Award is that it includes a lot of 11s and I'm wondering why....
5. I never win anything in contests. The one - and only - time I won anything of value was at a conference with my principal during my second year of teaching. I won a year-long license so our school could use some computerized reading program. The school got the program and I got a pat on the back. :-)
6. With two boys playing hockey, I spend a lot of time sitting in ice rinks. When I tell people that I wrote a lot of my dissertation at a hockey rink - I'm not kidding. I'd write during practices but watch during games.
7. My parents were high school sweethearts, my in-laws were high school sweethearts, and my husband and I are high school sweethearts. Given that family history, I've joked that if our boys are dating girls we don't like in high school, we should NOT consider it "just a phase".
8. My favorite vacation spot with my kids is Disney World. With my husband, it would definitely be Paris, France!
9. My favorite book is Goodnight Moon. Guess How Much I Love You is a close second.
10. I am terrified of heights and can't do rides that spin. I'm not really much fun at an amusement park but I'm happy to people-watch and hold your sunglasses, cell phone, etc.
11. Growing up, everyone in my family had the same initials - J.A.S.
Answers to Jacki's 11 Questions
1. Why did you become a teacher?
I love learning - school has been always been one of my favorite places. I am fortunate to learn so much from students every day!!!! Teaching is really just a continuation of learning.
2. If you couldn't have a job in education, what job would you choose?
I always wanted to be an editor for a publisher. Or a dolphin trainer at Sea World.
3. What is your favorite movie? Why?
This is tough - I used to love watching movies but haven't had much time since the kids have gotten involved in activities. Probably a tie between The Sound of Music and Disney's Sleeping Beauty for favorite movies when I was a kid.....
4. What is something that you want to do but you've never had the time, money, or chance to do it?
Spend a couple of months wandering through Europe.... We've made it to Paris but still so many more places to explore!
5. If you could have dinner with anyone, living or not living, who would it be? Why?
This is an easy one - my mom. She was such a huge influence on my life and I'd love the chance to fill her in on what has happened since her passing and thank her again for being such an amazing parent.
6. What is your earliest memory?
Reading with my mom.
7. What is one piece of advice that you would give a new teacher?
Keep balance in your life! It is easy to get wrapped up in all of the schoowork - it never ends! Be sure to enjoy activities outside of school. It will keep you well-rounded and help you make connections with your students and their families!
8. Who has been the most influential person in your life?
My mom.
9. What book are you currently reading?
Linchpin by Seth Godin and Agents of Change by Lucy West
10. What book are you planning to read next?
I'd like to get back to Blended Learning in Grades 4-12 by Caitlin Tucker. I started it a few months ago and keep getting distracted by other books.
11. What is your favorite ice cream flavor?
Right now, Jeni's Dark Chocolate Peppermint. But, when it's out of season, mint chocolate chip.
I've taken too long to write my Sunshine Award and all of the bloggers I know have already been nominated. Thanks, Jacki, for the opportunity to share!!!
Cathy Mere - another literacy guru in our district - describes the rules for the Sunshine Award on her blog (http://reflectandrefine.blogspot.com/):
The description of the Sunshine Award is shared by Matt Renwick:
The Sunshine Award gives others an opportunity to learn more about me as a blogger and then, in turn, I will send sunshine the way of 11 other amazing bloggers for you to get to know!
and the rules:
- Acknowledge the nominating blogger.
- Share 11 random facts about yourself.
- Answer the 11 questions the nominating blogger has created for you.
- List 11 bloggers. They should be bloggers you believe deserve some recognition and a little blogging love!
- Post 11 questions for the bloggers you nominate to answer and let all of the bloggers know they have been nominated. (You cannot nominate the blogger who nominated you.)
My first thought was that this will be MUCH more than I usually post - but I love the idea of sharing with other bloggers. So here it is....
11 Random Facts about Me:
1. Math was NOT my first love. I read voraciously as a child - and I still love to curl up with a good book. However, I enjoy teaching math more than I enjoy teaching language arts.
2. In college, I was the director of advertising for our student newspaper. Basically, I worked my way up the ladder after answering an ad to sit in the office to answer the phone. It was an easy way to earn a little spending money - and made me appreciate how much effort goes into creating a newspaper. I proofed the ads for our paper while several aspiring journalists were frantically trying to meet deadlines.
3. My husband and I were in the same 5th grade class. I like to remind students to be kind to everyone because you never know who you'll wind up with in life. We didn't talk in 5th grade but started dating in high school.
4. One of the things I noticed immediately about the Sunshine Award is that it includes a lot of 11s and I'm wondering why....
5. I never win anything in contests. The one - and only - time I won anything of value was at a conference with my principal during my second year of teaching. I won a year-long license so our school could use some computerized reading program. The school got the program and I got a pat on the back. :-)
6. With two boys playing hockey, I spend a lot of time sitting in ice rinks. When I tell people that I wrote a lot of my dissertation at a hockey rink - I'm not kidding. I'd write during practices but watch during games.
7. My parents were high school sweethearts, my in-laws were high school sweethearts, and my husband and I are high school sweethearts. Given that family history, I've joked that if our boys are dating girls we don't like in high school, we should NOT consider it "just a phase".
8. My favorite vacation spot with my kids is Disney World. With my husband, it would definitely be Paris, France!
9. My favorite book is Goodnight Moon. Guess How Much I Love You is a close second.
10. I am terrified of heights and can't do rides that spin. I'm not really much fun at an amusement park but I'm happy to people-watch and hold your sunglasses, cell phone, etc.
11. Growing up, everyone in my family had the same initials - J.A.S.
Answers to Jacki's 11 Questions
1. Why did you become a teacher?
I love learning - school has been always been one of my favorite places. I am fortunate to learn so much from students every day!!!! Teaching is really just a continuation of learning.
2. If you couldn't have a job in education, what job would you choose?
I always wanted to be an editor for a publisher. Or a dolphin trainer at Sea World.
3. What is your favorite movie? Why?
This is tough - I used to love watching movies but haven't had much time since the kids have gotten involved in activities. Probably a tie between The Sound of Music and Disney's Sleeping Beauty for favorite movies when I was a kid.....
4. What is something that you want to do but you've never had the time, money, or chance to do it?
Spend a couple of months wandering through Europe.... We've made it to Paris but still so many more places to explore!
5. If you could have dinner with anyone, living or not living, who would it be? Why?
This is an easy one - my mom. She was such a huge influence on my life and I'd love the chance to fill her in on what has happened since her passing and thank her again for being such an amazing parent.
6. What is your earliest memory?
Reading with my mom.
7. What is one piece of advice that you would give a new teacher?
Keep balance in your life! It is easy to get wrapped up in all of the schoowork - it never ends! Be sure to enjoy activities outside of school. It will keep you well-rounded and help you make connections with your students and their families!
8. Who has been the most influential person in your life?
My mom.
9. What book are you currently reading?
Linchpin by Seth Godin and Agents of Change by Lucy West
10. What book are you planning to read next?
I'd like to get back to Blended Learning in Grades 4-12 by Caitlin Tucker. I started it a few months ago and keep getting distracted by other books.
11. What is your favorite ice cream flavor?
Right now, Jeni's Dark Chocolate Peppermint. But, when it's out of season, mint chocolate chip.
I've taken too long to write my Sunshine Award and all of the bloggers I know have already been nominated. Thanks, Jacki, for the opportunity to share!!!
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