There are so many lessons available to teachers - it can be both a blessing and a curse. Finding different resources allows educators to personalize the learning - both for their students and themselves. How do you know whether the lessons are high-quality? What might work well in the lesson? What might need to be revised to better fit your needs? Rubrics might be one way you can use to help you preview lessons you find and determine how they might best fit your needs.
I have been so fortunate to work with Achieve's EQuIP (Educators Evaluating Quality Instructional Products) over the past year. Our group is comprised of 55 educators from across the country. We look at lessons for mathematics and English language arts. Using the EQuIP rubric, we review lessons and units, giving feedback to the developers about what works well within the lesson/unit and what might be improved. During our meeting in Washington D.C. this summer, the Teaching Channel came to record our work. You can see more about Achieve's work here: https://www.teachingchannel.org/videos/better-common-core-lessons-equip
If you are interested in using the rubric, here is a link to the EQuIP rubric:
http://www.achieve.org/files/EQuIPmathrubric-06-17-13_1.pdf (for mathematics)
http://www.achieve.org/files/EQuIP-ELArubric-06-24-13-FINAL.pdf (for ELA/literacy)
http://www.achieve.org/files/K-2ELALiteracyEQuIPRubric-07-18-13_1.pdf (for K-2 ELA/literacy)
Lessons and units that are judged to be "Exemplary" or "Exemplary if Improved" are shared on the EQuIP website and are free for anyone to use. When developers submit lessons to Achieve, they agree that their work will be shared with others if it receives the "E" or "E/I" rating. You can find those lessons here: http://www.achieve.org/EQuIP
Ohio has created a similar rubric, based on the EQuIP rubric, to help educators evaluate lessons they might use in their classrooms. You can find the Ohio Quality Review Rubric for mathematics here: https://education.ohio.gov/getattachment/Topics/Academic-Content-Standards/Mathematics/Resources-Ohio-s-New-Learning-Standards-K-12-Mathe/Mathematics-Quality-Review-Rubric.pdf.aspx
I hope that you will use some of these tools to talk with your colleagues about the lessons you are teaching. Many teachers no longer use textbooks and these rubrics are one of several available tools to help teachers know that they are using high-quality resources.
Sunday, August 24, 2014
Sunday, August 17, 2014
Seeing Through the Eyes of your Students
During my coursework at Ohio State, I read a powerful article that made me think more deeply about what questions we might ask students. The article was written by Shelly Harkness who is currently a professor of mathematics education at University of Cincinnati. Entitled "Social constructivism and the Believing Game: a mathematics teacher's practice and its implications', the article examined how teachers responded to students when the teacher took the position that what the student said was correct - regardless of the accuracy of the student's statement. You can access the full article here: http://bit.ly/1rgRAEY
Harkness writes of a student who responds "false" to the statement that All triangles have three sides. The student is asked to come to the board and draw a figure to represent what she sees. She draws the following:
The teacher responded, "No. That's not a triangle. It's not flat. The answer must be true." The conversation ended and the class moved on.
Harkness writes that the drawing above made her reconsider her own thinking about the mathematics. She wondered if Kayla looked at each side individually, thinking each of the four sides was flat. What questions might you ask Kayla to explore three-dimensional space? What discussion might you have about sides and faces? Or about geometric language?
What does Kayla know about triangles? How might this knowledge be influencing her interpretation of the question?
As we begin this new school year, I encourage you to assume all answers your students give are correct. At least from their point of view. Children do not intentionally give incorrect answers - the answer they give has validity from their point of view. Working to understand why they believe they are correct is the first step to uncovering their misconceptions and working to build their understanding. This school year, work to build meaning together WITH your students.
Harkness writes of a student who responds "false" to the statement that All triangles have three sides. The student is asked to come to the board and draw a figure to represent what she sees. She draws the following:
Harkness writes that the drawing above made her reconsider her own thinking about the mathematics. She wondered if Kayla looked at each side individually, thinking each of the four sides was flat. What questions might you ask Kayla to explore three-dimensional space? What discussion might you have about sides and faces? Or about geometric language?
What does Kayla know about triangles? How might this knowledge be influencing her interpretation of the question?
As we begin this new school year, I encourage you to assume all answers your students give are correct. At least from their point of view. Children do not intentionally give incorrect answers - the answer they give has validity from their point of view. Working to understand why they believe they are correct is the first step to uncovering their misconceptions and working to build their understanding. This school year, work to build meaning together WITH your students.
Monday, August 4, 2014
How I spent my summer vacation.....
We are getting ready to begin another school year - and I am excited about the opportunities I have had over the past few weeks that will influence my work for the upcoming year! It has been a busy summer and too long since I've written on this blog. Here are a few highlights:
- 4 days in Columbus, OH with the Network of Regional Leaders (NRLs) and the Ohio Department of Education. One day in June was spent debriefing the previous year with the Mathematics Cohort. Then we spent 3 days in July with NRLs from all disciplines at the Summer Leadership Academy.
- 1 1/2 days in Washington D.C. with EQuIP peer reviewers reviewing lessons for Achieve and sharing our work the the Teaching Channel.
- Many hours working on a parent website as a resource for parents in our district.
- Many more hours working on math resources for 5th grade teachers in our district.
- Meetings to plan professional development for using ST Math - our district is so fortunate to be part of a consortium that won a Straight A Fund grant from the State of Ohio to provide this opportunity for our students!
- Tutoring 2 wonderful girls who are going into 5th grade!!!
- Presenting at the Innovative Learning Environments 2014 conference in Hilliard, OH.
- Participating in Jo Boaler's "How to Learn Math" course through Standford University.
- Reviewing manuscripts for the Ohio Journal of School Mathematics.
- Reading great articles about teaching mathematics!!!
It wasn't ALL work! We did manage to get away for a family vacation and I enjoyed time with family and friends this summer. I feel so fortunate to have had these opportunities to spend time with so many talented educators! I hope to share what I have learned with you in the upcoming weeks.
I hope you have had the opportunity to recharge this summer - and that you have a WONDERFUL start to this school year!
Monday, April 14, 2014
Using Number Pieces for Work with Decimals
I've written before about Number Pieces - one of my favorite apps! (It is free - I hope you give it a try!) While there are obvious connections to the primary grades for this app, how might you use it in the intermediate grades?
One possibility is for addition and subtraction of decimals. Using base-10 blocks for decimals is a shift for students. Since early school experiences, many students have thought of the unit (or small cube) as "one". Before using base-10 blocks with decimals, it is important to give students opportunities to explore changing the meaning of "one". If "one" is the long/rod, what does the unit cube represent? If "one" is the flat, what does the long/rod represent?
After students have a chance to explore these relationships, the base-10 blocks can be a useful tool. For example, in the problem "Mary bought 9 pens. Each pen cost $0.35. How much did Mary spend on the pens?" students might use repeated addition to solve the problem. They might set up the problem to show that the flat will represent $1.00 and 3 tenths and 5 hundredths represent the cost of one pen.
Then students might draw 9 representations of 3 tenths and 5 hundredths to show the 9 pens. They can show the repeated addition as 0.35 x 9.
If you use the "lasso" tool, you can group all of the blocks together and then click the "join" button. This will group the tenths into ones, as well as the hundredths into tenths.
Using the "lasso" again, the app will join the tenths into ones. The student can then see that 0.35 x 9 is equal to 3.15.
This app provides students with a tool to visualize joining situations (and separating situations if you use the different colors) with decimals. This can be an engaging tool for students and help students create mental representations of the problems they are solving!
One possibility is for addition and subtraction of decimals. Using base-10 blocks for decimals is a shift for students. Since early school experiences, many students have thought of the unit (or small cube) as "one". Before using base-10 blocks with decimals, it is important to give students opportunities to explore changing the meaning of "one". If "one" is the long/rod, what does the unit cube represent? If "one" is the flat, what does the long/rod represent?
After students have a chance to explore these relationships, the base-10 blocks can be a useful tool. For example, in the problem "Mary bought 9 pens. Each pen cost $0.35. How much did Mary spend on the pens?" students might use repeated addition to solve the problem. They might set up the problem to show that the flat will represent $1.00 and 3 tenths and 5 hundredths represent the cost of one pen.
If you use the "lasso" tool, you can group all of the blocks together and then click the "join" button. This will group the tenths into ones, as well as the hundredths into tenths.
Using the "lasso" again, the app will join the tenths into ones. The student can then see that 0.35 x 9 is equal to 3.15.
This app provides students with a tool to visualize joining situations (and separating situations if you use the different colors) with decimals. This can be an engaging tool for students and help students create mental representations of the problems they are solving!
Tuesday, April 1, 2014
Learning Progressions and CCSS
In the past few weeks, several of my friends have shared examples that have been posted about math problems. A concern is that Common Core is changing the mathematics that we teach. Here is one example:
My first thought about this exercise is that the "old fashion" way is obviously simpler - especially if one already knows the traditional algorithm (or method). The "new" way is a form of counting on and takes more steps than the previous method.
I also notice that the problem is very simplistic - there is no need for regrouping in this example. Both methods are valid ways to solve the problem. Students that are fluent in using counting on would be able to solve this exercise quickly and efficiently by counting on.
Now, consider the problem 62 - 45. When regrouping is involved, this is a more complex problem. To use the "old fashion" way shown above, one would need to regroup from the 6 tens. Now there are 5 tens and 12 ones (or we've changed 60 + 2 to 50 + 12). Then one needs to subtract 12 - 5 in the ones place to get 7. Next, one needs to subtract 50 - 40 to get 10. So the answer is 17.
In using the "new" way, one could count up from 45 to 50 (adding 5), then from 50 to 60 (adding 10), and finally from 60 to 62 (adding 2). 5 + 10 + 2 is 17.
For students with strong number sense, the second method is quick and efficient. This method is also less prone to regrouping errors. Students should use a method that is efficient, mathematically valid, and generalizable according to Campbell, Rowan & Suarez (1998). This description can be applied to both of the solution methods shown in the above photo.
Another concern I have heard from parents is that Common Core does not teach the standard (or traditional) algorithms. A quick review of Common Core State Standards will demonstrate that this is NOT an accurate statement. In grade 4, students are expected to add and subtract multidigit whole numbers using the standard algorithm. In grade 5, students are expected to multiply multidigit whole numbers using the standard algorithm. In grade 6, students are expected to divide multidigit numbers using the standard algorithm.
So, what are students doing prior to grade 4? In the primary grades, students should be developing strong number sense and inventing strategies (student-invented, not teacher-demonstrated) to solve contextual problems. As students' understanding of place value deepens and strategies become more sophisticated, they will transition to expanded algorithms. These expanded algorithms reveal place value and what happens to numbers when we operate upon them. When students have a strong understanding of these expanded algorithms, it is a natural transition to move to the standard algorithms. Michael Battista writes extensively about these learning progressions in his work with Cognition-Based Assessment.
These progressions are based upon research about mathematics and how children learn. As educators, we need to do a better job helping parents understand why our teaching methods have changed and demonstrate that their children will be proficient mathematicians!
I also notice that the problem is very simplistic - there is no need for regrouping in this example. Both methods are valid ways to solve the problem. Students that are fluent in using counting on would be able to solve this exercise quickly and efficiently by counting on.
Now, consider the problem 62 - 45. When regrouping is involved, this is a more complex problem. To use the "old fashion" way shown above, one would need to regroup from the 6 tens. Now there are 5 tens and 12 ones (or we've changed 60 + 2 to 50 + 12). Then one needs to subtract 12 - 5 in the ones place to get 7. Next, one needs to subtract 50 - 40 to get 10. So the answer is 17.
In using the "new" way, one could count up from 45 to 50 (adding 5), then from 50 to 60 (adding 10), and finally from 60 to 62 (adding 2). 5 + 10 + 2 is 17.
For students with strong number sense, the second method is quick and efficient. This method is also less prone to regrouping errors. Students should use a method that is efficient, mathematically valid, and generalizable according to Campbell, Rowan & Suarez (1998). This description can be applied to both of the solution methods shown in the above photo.
Another concern I have heard from parents is that Common Core does not teach the standard (or traditional) algorithms. A quick review of Common Core State Standards will demonstrate that this is NOT an accurate statement. In grade 4, students are expected to add and subtract multidigit whole numbers using the standard algorithm. In grade 5, students are expected to multiply multidigit whole numbers using the standard algorithm. In grade 6, students are expected to divide multidigit numbers using the standard algorithm.
So, what are students doing prior to grade 4? In the primary grades, students should be developing strong number sense and inventing strategies (student-invented, not teacher-demonstrated) to solve contextual problems. As students' understanding of place value deepens and strategies become more sophisticated, they will transition to expanded algorithms. These expanded algorithms reveal place value and what happens to numbers when we operate upon them. When students have a strong understanding of these expanded algorithms, it is a natural transition to move to the standard algorithms. Michael Battista writes extensively about these learning progressions in his work with Cognition-Based Assessment.
These progressions are based upon research about mathematics and how children learn. As educators, we need to do a better job helping parents understand why our teaching methods have changed and demonstrate that their children will be proficient mathematicians!
Monday, March 10, 2014
Making equal parts
For this post, we're back to fractions....
The third grade classes are beginning to explore fractions. To introduce the idea of equal parts (an essential foundation for fractions), we worked on equal parts situations on-line using ST Math.
Students had to figure out which object to choose and then we discussed why our choice was correct. (If you choose correctly, the penguin is able to walk across the screen.) After conversation about what made the parts equal, students were given geoboards so they could explore equal parts with a partner.
These girls created parts that were not equal. They shared with me that the parts were not equal because they did not cover the same area. To find the area, the girls counted the "squares" between the pegs. (The third grade classes had just finished a study of area - so this was a great connection!)
These boys created parts that were equal. They described the parts as being the "same". The boys in this group are learning English as a second language and this activity gave them a great opportunity to explore examples and nonexamples of equal parts.
After having time to explore with their partner, we came back together as a whole group. Students were asked to share one of their representations and explain whether the parts were equal or not - and how they knew. This group described knowing the parts were equal because they divided the square into four equal parts (smaller squares). Then they divided each square in half. Such great conversation and thinking from these third grade students!
As a challenge to the class, I left them with three representations on geoboards. Each geoboard was divided into fourths - one with squares, one with rectangles, and one with triangles. Students agreed that each of the geoboards was divided into equal parts. I then asked whether the triangle was equal to the square - or if the rectangle was equal to the square. Students were intrigued by this idea - intuitively they thought yes but they weren't certain. I left the problem as an open question and asked them to think about it further. I can't wait to hear their ideas!
The third grade classes are beginning to explore fractions. To introduce the idea of equal parts (an essential foundation for fractions), we worked on equal parts situations on-line using ST Math.
As a challenge to the class, I left them with three representations on geoboards. Each geoboard was divided into fourths - one with squares, one with rectangles, and one with triangles. Students agreed that each of the geoboards was divided into equal parts. I then asked whether the triangle was equal to the square - or if the rectangle was equal to the square. Students were intrigued by this idea - intuitively they thought yes but they weren't certain. I left the problem as an open question and asked them to think about it further. I can't wait to hear their ideas!
Sunday, March 2, 2014
#GETMath: March 2014
I have posted a new #GETMath task for March 2014. http://bit.ly/1fA0GGf This task is based on Goldbach's Conjecture. I selected this task because there are some similarities to the Sums of Consecutive Numbers task from February. It might be interesting to discuss the ways in which the two problems are alike and how they differ.
This task also has multiple entry points. Students are adding two numbers - which students begin to do in grade 1. For students who might struggle with computation, you can give them a calculator. For students who might not know the prime numbers, you can give them a list of the prime numbers.
For students who need an extension, they can explore finding prime numbers and how one might generate that list. These students could also work to generalize or write a rule for the patterns.
Remember that the purpose of these tasks is not necessarily to focus on content. While mathematical content is important, these tasks lend themselves to the Standards for Mathematical Practice - an essential part of mathematical learning. Use these as an opportunity to engage your students in mathematics and get them excited about mathematics so they are eager to learn additional content!
I hope that you will share your thinking and pictures on Twitter! I look forward to seeing what you have to share!
This task also has multiple entry points. Students are adding two numbers - which students begin to do in grade 1. For students who might struggle with computation, you can give them a calculator. For students who might not know the prime numbers, you can give them a list of the prime numbers.
For students who need an extension, they can explore finding prime numbers and how one might generate that list. These students could also work to generalize or write a rule for the patterns.
Remember that the purpose of these tasks is not necessarily to focus on content. While mathematical content is important, these tasks lend themselves to the Standards for Mathematical Practice - an essential part of mathematical learning. Use these as an opportunity to engage your students in mathematics and get them excited about mathematics so they are eager to learn additional content!
I hope that you will share your thinking and pictures on Twitter! I look forward to seeing what you have to share!
Chatting about Sums of Consecutive Numbers
It has been so much fun to connect with classes about the Sums of Consecutive Numbers problem! I have been so impressed with the thinking and problem solving skills of students! A 1st grade class had the fantastic idea to use a measurement tool as a number line!
A 4th grade class at my school used video chat to connect with a 4th grade class in Dublin, Ohio. Before our chat, students explored the problem and discussed questions with their homeroom classes. Some of the questions they explored included:
A 4th grade class at my school used video chat to connect with a 4th grade class in Dublin, Ohio. Before our chat, students explored the problem and discussed questions with their homeroom classes. Some of the questions they explored included:
- Were there any numbers that couldn't be made by the sum of consecutive numbers?
- What happened if you added 3 numbers? Or 4 numbers?
- Could you predict whether the sum would be even or odd based on the number of addends?
- Is there a pattern? If so, how might you describe it?
- Is there anything special about the prime numbers in comparison to the composite numbers? What do you notice?
Last week, the classes were able to connect and share their observations:
Some of the observations students made included:
- When you have 2 consecutive addends, the sum is an odd number. For example, 3 + 4 = 7.
- When you have 3 consecutive addends, the sum is a multiple of 3. For example, 1 + 2 + 3 = 6 and 6 is a multiple of 3.
- When you have 4 consecutive addends, the sum is NOT a multiple of 4. They are curious why the pattern that seems to work for 3 addends does not work with 4 addends. For example, 1 + 2 + 3 + 4 = 10 and 10 is not a multiple of 4.
- When you have 5 consecutive addends, the sum is a multiple of 5. For example, 1 + 2 + 3 + 4 + 5 = 15 and 15 is a multiple of 5.
The conversation generated further questions. We're hoping that students can explore these questions in groups and then we can have small groups video chat rather than a whole group video chat.
Using technology to connect with others and share our mathematical thinking is a great way to take our learning beyond the physical walls of our school building!
Tuesday, February 11, 2014
#GETMath: Launching the Problem for Sums of Consecutive Numbers
Have you had a chance to try the Sums of Consecutive Numbers problem with your students? If not, you might be wondering how you might launch the problem. I did not give specific ways to launch the problem in the video because of the wide range of learners that might be participating. After further reflection, I thought it might be worthwhile to share how I involved students in the problem.
I started the problem with a grade 3 class late last week. We began by talking about the Global Read Aloud project and what they enjoyed about it. Students said what they enjoyed most was talking with others about the book. After explaining that I wanted to do something similar with a math problem, the class was eager to get started! I shared the title of the problem "Sums of Consecutive Numbers" and asked students to tell me what they knew about "sums". The class quickly agreed that sums were answers to addition problems. Defining "consecutive" was a little more challenging but with the help of a dictionary, students soon agreed that consecutive numbers would be numbers that follow one another in sequence, such as on a number line.
After drawing a number line to use as a reference, we began to brainstorm possible answers. On a large chart, students were able to write their answers once they shared their idea and the class was in agreement that the example fit the rules of the problem.
After we brainstormed a few examples and I felt that students had a strong understanding of the problem, I encouraged students to explore the numbers on their own. Students received a chart for them to record their thinking individually. To begin, students were encouraged to copy the examples from the class chart.
While students were working, I circulated the room and asked questions about their strategies. Some students immediately saw a pattern with the odd numbers - noticing that any odd number could be made as the sum of two consecutive numbers. Other students tried using three or four addends and writing down the sum they discovered.
Students worked for about 15 minutes and then I stopped the class. We discussed our observations about the patterns and wrote them on a chart. I encouraged students to continue working on the problem and told them that they should not be limited to the 50 numbers on the chart. I asked students to continue adding to the chart as they noticed other patterns. I can't wait to check in this week to see what else they have added to the chart!
If you have been waiting for the moment to introduce this problem, I hope this helps you get started! The students were so engaged with this problem that they continued working during their indoor recess! Please ask questions or share how things are going on Twitter at #GETMath! I look forward to hearing from you!
I started the problem with a grade 3 class late last week. We began by talking about the Global Read Aloud project and what they enjoyed about it. Students said what they enjoyed most was talking with others about the book. After explaining that I wanted to do something similar with a math problem, the class was eager to get started! I shared the title of the problem "Sums of Consecutive Numbers" and asked students to tell me what they knew about "sums". The class quickly agreed that sums were answers to addition problems. Defining "consecutive" was a little more challenging but with the help of a dictionary, students soon agreed that consecutive numbers would be numbers that follow one another in sequence, such as on a number line.
Wednesday, February 5, 2014
Using #GETMath to teach the Standards for Mathematical Practice
Some of you may have thought about using the Sums of Consecutive Numbers and joining in the #GETMath conversation but you're feeling stuck because you don't feel like the problem clearly connects to the content for your grade level. (If you haven't heard about #GETMath, learn more here: http://bit.ly/1eybFKx)
This problem does not need to be (nor should it necessarily be) completed in one or two class periods. Students can play with the problem during the month of February and share their observations once or twice each week.
This is an opportunity emphasize the Standards for Mathematical Practice!
- Make sense of problems and persevere in solving them.
- Sums of Consecutive Numbers has many different aspects to explore. In introducing the problem to a 4th grade class, some students immediately thought to use more than two addends. Another student made a conjecture that no even number could be made by only two addends. The students were ready to get started and excited to look for patterns! We created a chart where they can confirm or revise their findings and add additional observations.
- Reason abstractly and quantitatively.
- Because this standard focuses on connecting real situations with symbolic representations, it is not really a standard that could be emphasized during this exploration.
- Construct viable arguments and critique the reasoning of others.
- As noted above, students were making conjectures about the patterns and working to prove or disprove their thinking. As students continue working through the problem, they can add to the class chart. We even plan to video chat with another 4th grade class - what a great opportunity to share our arguments and discuss our thinking about the problem!
- Model with mathematics.
- As with SMP #2, this would not be the best standard to apply with the Sums of Consecutive Numbers problem. Students might use a table to organize their thinking but they do not apply math to solve a real-world problem.
- Use appropriate tools strategically.
- As with SMP #2 and #4, this would not be a strong standard to emphasize with this particular problem. While the table or organizer used by students might be considered a tool, it would be a weak connection to this standard.
- Attend to precision.
- In the Sums of Consecutive Numbers problem, students need to be sure that the numbers they use are indeed consecutive numbers and check to be sure the sum they find is accurate. They also need to be able to communicate their thinking in a way that is clearly understood by others. This would be a strong standard to emphasize with this problem!
- Look for and make use of structure.
- The Sums of Consecutive Numbers problem provides students with an opportunity to notice and discuss the structure of mathematics. For example, why can the odd numbers always be made by the sums of two consecutive numbers?
- Look for and express regularity in repeated reasoning.
- The Sums of Consecutive Numbers problem provides students with an opportunity to observe patterns and reason about generalizations. For example, what numbers cannot be made by the sums of consecutive numbers? Could you predict the next number that cannot be made as a sum of consecutive numbers? Can you write a rule that will help you find all of the numbers that cannot be made by the sum of consecutive numbers?
The Sums of Consecutive Numbers problem has clear connections to five of the eight Standards for Mathematical Practice. The CCSS emphasizes that the Standards for Mathematical Practice are as essential as the mathematical content. While it would be difficult to emphasize five Mathematical Practices within your lesson, there are many opportunities for you to engage students in the Standards for Mathematical Practice through this problem.
I hope you will join us and share your thinking at #GETMath!
Wednesday, January 29, 2014
#GETMath
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
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
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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