Solve the following problem using two different methods.
34 - 9
Share your methods in the comments below.
Friday, April 29, 2016
Wednesday, April 27, 2016
Curriculum Issues
I taught for two years with no centralized curriculum or standards-aligned textbooks to use. While many veteran teachers might love this amount of freedom, it was terrifying for a new teacher. I was able to make it work by focusing on the standards and pulling resources from multiple sources to create something usable for my students, but finally found a free curriculum to use my third year in the form of EngageNY. While it was not the end all be all answer, it was a wonderful place for me to start. Because I'd focused so heavily on my state's standards, I was able to modify lessons and units within EngageNY to fit those standards as well as meet the levels of my students and my own teaching style.
I highly recommend finding some sort of curriculum to use as a foundation for your lesson planning. I say this with a word of warning, however. Textbook and curriculum publishers have been notorious for placing whatever sticker they need to on the cover of their publication and selling it to states. This means that just because it says "Common Core Aligned," or "TEKS Aligned," etc., does not necessarily mean it is. [You can check out EdReports for an independent review of educational materials.]
Even if it is properly aligned, often the material isn't rigorous enough to be used without any modifications. Sometimes the questions are posed as "higher order thinking questions," but provide too much information to the students, and end up being a simple substitution problem. Other times, it is marked as a "modeling problem," but, again, all of the information is provided for the students. Here is a great example from Dan Meyer's blog:
Mathematical modeling is defined in similar terms by the Common Core State Standards, the modeling cycle, or the IB:
In the above problem:
This does not have to be the awful question it appears to be. This is asking students to decompose numbers and then regroup. For example, the student can rewrite the problem to be 8+2+3, which would result in 10+3. Or the student can rewrite the problem to be 3+5+5, which would result in 3+10. This is a bad question because of how it's asked, but not because of the underlying math the problem is trying to get at.
The standards are not a curriculum. They are standards students are expected to meet to help to close achievement gaps and prepare them for college and the workforce. The way teachers (or schools or districts) put in place curriculum and meet those standards is entirely their own, so there is no such thing as a "Common Core math problem," just like there is no such thing as a "TEKS math problem."
It is our job to make sure the questions we're asking our students align to the standards of our states and help our students master the skills outlined in those standards. If the textbook or curriculum we have been directed to use (or have chosen to use) does not meet that criteria, it is our job to modify when necessary.
What challenges have you faced with your curriculum?
I highly recommend finding some sort of curriculum to use as a foundation for your lesson planning. I say this with a word of warning, however. Textbook and curriculum publishers have been notorious for placing whatever sticker they need to on the cover of their publication and selling it to states. This means that just because it says "Common Core Aligned," or "TEKS Aligned," etc., does not necessarily mean it is. [You can check out EdReports for an independent review of educational materials.]
Even if it is properly aligned, often the material isn't rigorous enough to be used without any modifications. Sometimes the questions are posed as "higher order thinking questions," but provide too much information to the students, and end up being a simple substitution problem. Other times, it is marked as a "modeling problem," but, again, all of the information is provided for the students. Here is a great example from Dan Meyer's blog:
Mathematical modeling is defined in similar terms by the Common Core State Standards, the modeling cycle, or the IB:
- identifying variables in the situation and selecting those that represent essential features,
- formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,
- analyzing and performing operations on these relationships to draw conclusions,
- interpreting the results of the mathematics in terms of the original situation,
- validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable,
- reporting on the conclusions and the reasoning behind them.
In the above problem:
- Who is identifying essential variables? Where?
- Who is formulating the model for those variables? Where?
- Etc.
The standards are not a curriculum. They are standards students are expected to meet to help to close achievement gaps and prepare them for college and the workforce. The way teachers (or schools or districts) put in place curriculum and meet those standards is entirely their own, so there is no such thing as a "Common Core math problem," just like there is no such thing as a "TEKS math problem."
It is our job to make sure the questions we're asking our students align to the standards of our states and help our students master the skills outlined in those standards. If the textbook or curriculum we have been directed to use (or have chosen to use) does not meet that criteria, it is our job to modify when necessary.
What challenges have you faced with your curriculum?
Monday, April 25, 2016
Number Talk Discussion #3
There are multiple ways to see the nine blocks in this figure. Below are a few of those ways:
Did you see 9 in a different way? Share below in the comments!
This Friday, I'll show you a different way to approach Number Talks using actual numbers, so be sure to join me for that!
Friday, April 22, 2016
Friday Number Talk #3
Without counting one by one, figure out how many boxes there are.
Share your method in the comments below.
Wednesday, April 20, 2016
The Anatomy of a Lesson
The anatomy of a lesson has the same basic structure whether you're teaching a block schedule or a traditional schedule. Today we are going to break down the lesson into it's key components.
Lesson Planning
Planning is crucial for both types of schedules. If you have a limited time with the students, every minute must be used efficiently and purposefully. Everything from passing out materials to getting into groups must have a procedure attached to it to avoid wasted time. If you have more time with students, you still need to make sure you are using your time efficiently. If you are switching from one type of schedule to another, I highly recommend creating lesson plans that are overly detailed until you become comfortable with the new setup. Include every activity, your estimated time for each activity, how you will transition from one thing to another, and questions you plan on asking your students as well as questions you anticipate your students asking you. Once you begin to get a feel for how long everything takes, you can always scale back on your lesson plans, but it is always better to be over-prepared than under-prepared.
Set the Timer
Whether you're teaching a block schedule or a traditional (50-60 minute) class, time management is key. This is often an area where novice teachers struggle the most, but even those of us who have been in the classroom for years still have issues with timing now and again. While teaching block might seem wonderful at first (90 whole minutes to teach! They're going to learn all the things!), often teachers struggle to fill that time. They tend to allow students more down time than a teacher in a 50 minute class would, and lose out on the benefit of having the students in their classroom longer. One of the easiest things to do is time everything from the bell ringer/do now/warm-up to the instructional portion of class to independent practice to the exit ticket/closure. This will not only help your students learn how to manage their own time while working on problems, but also help keep you honest and on track. Additionally, our students have a limited attention span (about 1 minute per year of life on this planet), and changing activities frequently helps keep students engaged. Some teachers think that when they change activities it has to be something big and dramatic, but often something simple like a quick think-pair-share moment is enough.
Variety
Scheduled and structured movement around the room can be a useful way to keep the classroom moving. You can set up Math Stations, a Gallery Walk, or simply have the students change seats to take notes. Block schedules also allow for a more thorough release of responsibility to students. There is time for Direct Instruction (I do), Group Practice (We do), then Independent Practice (You do). While traditional class periods allow for all of these, teachers often have to sacrifice the amount of time spent during group practice or independent practice due to time constraints.
Transitions
All transitions from one activity to another need to be smooth, logical, and clear to students, otherwise you will lose some of them along the way. If students need to put materials away before the next activity can begin, or get materials out, assign that job to a few students. Do not give important information to students during this time. Instead, make sure that students are doing what they need to do (moving desks, passing materials forward, etc.) in order to get ready for the next thing.
Review and Closure
It is imperative that teachers leave some time at the end of class for a review of the topics or skills learned in class. This helps students bring things together in their own mind and to conceptualize what has been taught. Closure activities can be questions asked by the teacher, a think-pair-share activity, or exit tickets. Exit tickets are an excellent way to check for understanding at the end of the lesson, and can be used as a formative assessment for the lesson.
All of these are important components of any good lesson. The trick to make it a great lesson is consistency and practice. If you feel like you are struggling with any of these, observe another teacher and take notes on what they do. Even if they don't teach your grade level or your content area, you can always learn from other teachers. Best practices are best practices no matter the subject.
What is the anatomy of your lessons?
Lesson Planning
Planning is crucial for both types of schedules. If you have a limited time with the students, every minute must be used efficiently and purposefully. Everything from passing out materials to getting into groups must have a procedure attached to it to avoid wasted time. If you have more time with students, you still need to make sure you are using your time efficiently. If you are switching from one type of schedule to another, I highly recommend creating lesson plans that are overly detailed until you become comfortable with the new setup. Include every activity, your estimated time for each activity, how you will transition from one thing to another, and questions you plan on asking your students as well as questions you anticipate your students asking you. Once you begin to get a feel for how long everything takes, you can always scale back on your lesson plans, but it is always better to be over-prepared than under-prepared.
Set the Timer
Whether you're teaching a block schedule or a traditional (50-60 minute) class, time management is key. This is often an area where novice teachers struggle the most, but even those of us who have been in the classroom for years still have issues with timing now and again. While teaching block might seem wonderful at first (90 whole minutes to teach! They're going to learn all the things!), often teachers struggle to fill that time. They tend to allow students more down time than a teacher in a 50 minute class would, and lose out on the benefit of having the students in their classroom longer. One of the easiest things to do is time everything from the bell ringer/do now/warm-up to the instructional portion of class to independent practice to the exit ticket/closure. This will not only help your students learn how to manage their own time while working on problems, but also help keep you honest and on track. Additionally, our students have a limited attention span (about 1 minute per year of life on this planet), and changing activities frequently helps keep students engaged. Some teachers think that when they change activities it has to be something big and dramatic, but often something simple like a quick think-pair-share moment is enough.
Variety
Scheduled and structured movement around the room can be a useful way to keep the classroom moving. You can set up Math Stations, a Gallery Walk, or simply have the students change seats to take notes. Block schedules also allow for a more thorough release of responsibility to students. There is time for Direct Instruction (I do), Group Practice (We do), then Independent Practice (You do). While traditional class periods allow for all of these, teachers often have to sacrifice the amount of time spent during group practice or independent practice due to time constraints.
Transitions
All transitions from one activity to another need to be smooth, logical, and clear to students, otherwise you will lose some of them along the way. If students need to put materials away before the next activity can begin, or get materials out, assign that job to a few students. Do not give important information to students during this time. Instead, make sure that students are doing what they need to do (moving desks, passing materials forward, etc.) in order to get ready for the next thing.
Review and Closure
It is imperative that teachers leave some time at the end of class for a review of the topics or skills learned in class. This helps students bring things together in their own mind and to conceptualize what has been taught. Closure activities can be questions asked by the teacher, a think-pair-share activity, or exit tickets. Exit tickets are an excellent way to check for understanding at the end of the lesson, and can be used as a formative assessment for the lesson.
All of these are important components of any good lesson. The trick to make it a great lesson is consistency and practice. If you feel like you are struggling with any of these, observe another teacher and take notes on what they do. Even if they don't teach your grade level or your content area, you can always learn from other teachers. Best practices are best practices no matter the subject.
What is the anatomy of your lessons?
Monday, April 18, 2016
Number Talks Discussion #2
This is an image of a Ten Frame. These are becoming more commonly used in the lower elementary grades to help students learn how to subitize (the ability to quickly identify the number of items in a small set without counting), gain number sense, and learn about place value.
Once students become familiar with Ten Frames, they can see that this particular image shows "2 less than 10," or "5+3," or "6+2." All of these are different ways to say "8."
Did you see "8" in a different way? Share your thoughts in the comments below!
Friday, April 15, 2016
Friday Number Talk #2
Without counting one by one, figure out how many dots there are.
Share your method in the comments below.
Wednesday, April 13, 2016
The Problem with Key Words
Key words are taught in math classes around the country. Teachers have anchor charts posted around the room with "helpful" reminders that "in all" means to add and that "left" indicates you should subtract. This method has been reinforced by the overly simple and formulaic story problems (word problems for those of us who are old school) often found in textbooks. We often present the key word approach to problem solving with good intentions. We want our students to know when certain operations should be used, and we think the best way to help them understand is to provide words that are linked to those operations. In theory, this works. In practice, however, we quickly realize there are limitations to the magic of key words.When students rely on key words, they cease to reason about or understand the situation posed within the story problem. The key word approach encourages children to ignore the meaning and structure of the problem and look for an easy way out. This is why our students end up using the incorrect operations and are inconsistent with their mastery of solving word problems. In short, key words are the bane of my teachering experience.
Let's examine a story problem together.
I had 5 apples in my basket on Monday. On Tuesday I increased the amount of apples so now I have 7 altogether. How many apples did I add on Tuesday?
If the student is using the key-word approach, "increased," "altogether," and "add" would indicate they need to add 5 and 7 together to get 12 apples. The key words here are misleading, and will not help our students understand or solve this problem.
So how do we approach story problems if key words are not the answer? Building models is an excellent alternative! These can be physical models built with manipulatives or graphics that the students draw.
The example to the right is a bar model, and can be created with strips of paper or drawn as it is here. By comparing the part (the 5 apples I have on Monday) to the whole (the 7 apples I now have on Tuesday) I understand I am adding onto 5 until I get to 7.The example below models the same problem, but uses manipulatives instead. We start with 5 apples, add some amount, and end up with 7 apples. From there, we can determine the number of apples needed to complete the problem.
Both of these methods allow students to start making connections between addition and subtraction and they can start to see how they are the inverses of each other. Both of these examples also help lay the foundation for Algebra later on, and both methods can be extended to help students understand multiplication and division and the relationship they have to one another. This is something that simply using key words does not help our students understand.
Here are some questions that you can use in whole class discussions or have students answer on paper. Similar questions can be used for most problems:
- What is happening in this problem?
- What will the answer tell us?
- Do you think it will be a big number or a small number?
- What operation do you think you'll need to use to solve this problem? Why?
Requiring students to provide an explanation of their process is also important. The explanation can be an illustration or written sentences. If students choose to draw pictures, make sure you have a discussion with them about what makes a good drawing in mathematics: simple is good. While we want to encourage our students to be artistic, we do not want our students to lose sight of the mathematics they are trying to model.
How do you approach teaching word problems in your classroom? What has your experience been with your students using key words?
Saturday, April 9, 2016
A Discussion About Homework
Homework is a hot button issue for parents, students, teachers, and administration.
Well, I've looked at the research, and the results may surprise you! I'm not going to address all of the above questions in this post, but I'm including them because I think they're important for us to at least consider. [I'll link to the studies throughout the post in case you'd like to read them yourself.]
Elementary School
Most of the research shows that homework for elementary students is ineffective and actually detrimental to student achievement. Younger students have developed less effective study habits than older students. Many teachers assign homework for elementary students in order to help them learn to manage their time more effectively, but because younger students are less able to tune out distractions, it ends up being a test of how well the parents can manage the child's time. This isn't to say that homework should not be assigned to elementary students, but that we need to be more mindful of how much and exactly what we are assigning to those students.
Middle School and High School
The research shows homework is more effective for students in middle school and high school, with the biggest benefits being found in high school. There isn't necessarily a connection between homework and better grades, but there is one between homework and higher standardized test scores (which is a whole other can of worms).
How much time should the homework take?
So, how much homework should we be assigning? The research shows that 10 minutes in first grade is a good place to start. Every grade thereafter should add an additional 10 minutes, meaning second graders would get 20 minutes, and seniors in high school would get 120 minutes. This makes sense to me, since children's attention spans tend to get longer as they age.
What kind of homework should be assigned?
Now that we know about how much time our students should be spending on their homework, let's focus on the quality of homework we're assigning. If homework is not completed, it's not helping anyone (and if you're a teacher with limited copies or who has to buy their own paper, it's actually hurting someone!), so assigning homework does no good if students don't do it. Busy work turns students off from learning. If they can see the connection between what they're doing as homework and what they need to know for class, they are much more willing to do the homework. Students should not be learning new skills through homework. They should be practicing learned skills to help reinforce what they've learned in class. We have to make the homework we assign short, to the point, and purposeful. So, how do we do that?
Elementary Homework
Homework for elementary students needs to be designed in a way to help reinforce the child's natural love of learning and help students start to develop good study habits. You could assign 10 minutes of reading a book of the child's choice a night. This is a great way to involve parents or other family members in a non-threatening way. Again, students should be practicing skills already learned in class. This way, homework is not a struggle for the students (Or parents. How many times have teachers heard that the parents struggled to help their third grader with their homework!?!), but a time to practice learned skills. A sheet of math problems isn't the best way to practice skills. Instead, give them an assignment that requires them to find objects in their home and model different skills (addition, subtraction, multiplication, division) and illustrate what they did. This will help them see math in their every day lives.
Middle School and High School Homework
Once students reach these grades, teachers can start giving longer homework assignments, but they should still be purposeful. Students should be practicing a skill or process that students can do independently but not fluently, elaborating on information that has been addressed in class to deepen students' knowledge, and providing opportunities for students to explore topics of their own interest. Additionally, homework should be designed to maximize the chances that students will complete it. For example, ensure that homework is at the appropriate level of difficulty. Students in these grades should be able to complete homework assignments independently with relatively high success rates, but they should still find the assignments challenging enough to be interesting.
Grading Homework
Homework should be viewed as a formative assessment. It is an assignment that students are completing as they are learning the desired skills, and are therefore expected to make mistakes. Grading based on perfection penalizes students who do not grasp the concept completely on the first go-round, and typically deters students from even attempting in the future. However, simply grading based on completion doesn't allow for teachers to give feedback on skills or concepts the students are struggling with. The best approach seems to be to combine the two methods.
Let's say you assign 6 or 7 problems for the students to complete. Take off 5 points for each problem that is incorrect, but valiantly attempted and 15 points off for each problem that isn't attempted at all. Here's what their grades will look like if they at least give a good attempt at each problem:
Even if a student gets all of the problems incorrect, if they've at least given them their best effort, the student's grade won't be tanked. This also allows for teachers to provide fast and personalized feedback on the incorrect problems. Students who know their teacher will provide this kind of feedback are more likely to complete their homework more consistently.
- How much is too much for children to be doing a night? How much help should parents be giving their children on homework? If children never have homework, do parents think their child's teacher isn't doing their job properly?
- How should students approach homework? How can students develop good study habits and time management if they aren't given nightly homework?
- How much homework should those of us teachering assign? What should that homework look like if we are assigning it? And how should we grade it once we've assigned it?
- Should administrators implement a homework policy for their school? And what should that policy look like?
Well, I've looked at the research, and the results may surprise you! I'm not going to address all of the above questions in this post, but I'm including them because I think they're important for us to at least consider. [I'll link to the studies throughout the post in case you'd like to read them yourself.]
Elementary School
Most of the research shows that homework for elementary students is ineffective and actually detrimental to student achievement. Younger students have developed less effective study habits than older students. Many teachers assign homework for elementary students in order to help them learn to manage their time more effectively, but because younger students are less able to tune out distractions, it ends up being a test of how well the parents can manage the child's time. This isn't to say that homework should not be assigned to elementary students, but that we need to be more mindful of how much and exactly what we are assigning to those students.
Middle School and High School
The research shows homework is more effective for students in middle school and high school, with the biggest benefits being found in high school. There isn't necessarily a connection between homework and better grades, but there is one between homework and higher standardized test scores (which is a whole other can of worms).
How much time should the homework take?
So, how much homework should we be assigning? The research shows that 10 minutes in first grade is a good place to start. Every grade thereafter should add an additional 10 minutes, meaning second graders would get 20 minutes, and seniors in high school would get 120 minutes. This makes sense to me, since children's attention spans tend to get longer as they age.
What kind of homework should be assigned?
Now that we know about how much time our students should be spending on their homework, let's focus on the quality of homework we're assigning. If homework is not completed, it's not helping anyone (and if you're a teacher with limited copies or who has to buy their own paper, it's actually hurting someone!), so assigning homework does no good if students don't do it. Busy work turns students off from learning. If they can see the connection between what they're doing as homework and what they need to know for class, they are much more willing to do the homework. Students should not be learning new skills through homework. They should be practicing learned skills to help reinforce what they've learned in class. We have to make the homework we assign short, to the point, and purposeful. So, how do we do that?
Elementary Homework
Homework for elementary students needs to be designed in a way to help reinforce the child's natural love of learning and help students start to develop good study habits. You could assign 10 minutes of reading a book of the child's choice a night. This is a great way to involve parents or other family members in a non-threatening way. Again, students should be practicing skills already learned in class. This way, homework is not a struggle for the students (Or parents. How many times have teachers heard that the parents struggled to help their third grader with their homework!?!), but a time to practice learned skills. A sheet of math problems isn't the best way to practice skills. Instead, give them an assignment that requires them to find objects in their home and model different skills (addition, subtraction, multiplication, division) and illustrate what they did. This will help them see math in their every day lives.
Middle School and High School Homework
Once students reach these grades, teachers can start giving longer homework assignments, but they should still be purposeful. Students should be practicing a skill or process that students can do independently but not fluently, elaborating on information that has been addressed in class to deepen students' knowledge, and providing opportunities for students to explore topics of their own interest. Additionally, homework should be designed to maximize the chances that students will complete it. For example, ensure that homework is at the appropriate level of difficulty. Students in these grades should be able to complete homework assignments independently with relatively high success rates, but they should still find the assignments challenging enough to be interesting.
Grading Homework
Homework should be viewed as a formative assessment. It is an assignment that students are completing as they are learning the desired skills, and are therefore expected to make mistakes. Grading based on perfection penalizes students who do not grasp the concept completely on the first go-round, and typically deters students from even attempting in the future. However, simply grading based on completion doesn't allow for teachers to give feedback on skills or concepts the students are struggling with. The best approach seems to be to combine the two methods.
Let's say you assign 6 or 7 problems for the students to complete. Take off 5 points for each problem that is incorrect, but valiantly attempted and 15 points off for each problem that isn't attempted at all. Here's what their grades will look like if they at least give a good attempt at each problem:
-0 100%
|
-1 95%
|
-2 90%
|
-3 85%
|
-4 80%
|
-5 75%
|
-6 70%
|
-7 65%
|
Even if a student gets all of the problems incorrect, if they've at least given them their best effort, the student's grade won't be tanked. This also allows for teachers to provide fast and personalized feedback on the incorrect problems. Students who know their teacher will provide this kind of feedback are more likely to complete their homework more consistently.
What are your thoughts on homework?
Sunday, April 3, 2016
Using Manipulatives in the Classroom
One of the hardest things about teaching math to children is the fact that you're an adult. We don't think about numbers in the same way our students do because we've learned Algebra. Most of us were never taught using math manipulatives, or if we were, it was only in early elementary. Most of us were not taught in our college courses how to incorporate them into our teaching. We might have them in our classroom cabinet gathering dust because we're either afraid they will become weapons of mass chaos or we have no idea what to do with them. After this blog post, I hope your views on manipulatives will have changed and you'll be inspired to pull them out of the dark and use them with your students!
Why are Manipulatives Important?
Manipulatives help make math more concrete for students. Numbers always relate back to some amount of things, and allowing students to move those things around will help them understand what is happening when we apply different operations to them. Now, I know that most state tests do not allow for use of physical manipulatives, but students don't start the beginning of the year ready for those state tests. These are tools to help students build conceptual understanding of the math. Worst case scenario, the students use their scratch paper to draw representations of the manipulatives they used in the classroom throughout the year. Manipulatives also make math more engaging for students, and help them build their mathematical confidence. I know that in the beginning, it can seem like you're spending a lot of time trying to set expectations about how they should be used in class and trying to help students make connections between the math and the object, but in the long run the use of manipulatives in your classroom will cut down on the amount of remediation and re-teaching you have to do later on. This is because the students' understanding of the math will be more complete once they (and you!) get the hang of using the manipulatives in the classroom!
Classroom Management
Let's get the big scary stuff out of the way first. Children have a natural inclination towards playing. This is fine, as long as we are able to pull them back to the reality of the lesson and get them working. It's even better if we can trick them into thinking they're playing the whole time, while we're secretly teaching them. When you first give manipulatives to your students, allow them time to examine them and explore what they can do. You can structure this by including a question or two on their recording sheets asking them to describe the manipulative or write down things they observe.
Some students need a little help with organization, so providing a workspace for them is often very beneficial. This can be a piece of construction paper on their desk that their manipulatives need to stay on. You can extend this and require only the manipulatives that are part of their answer to be on the paper while the rest are on their desk. This will help you see what they are seeing, as well as keep things a little more orderly.
Some teachers have their students put both hands on top of their heads when it's time to listen to instructions. This will only work if you sell it to them, but even middle school students will buy into it if you believe in it. Having some sort of attention grabber and signal from the students is important, but make sure it fits with your personality and the personality of the class.
If the manipulatives end up in the air or on the floor in manners that are unacceptable, give a warning to the offender, and make it very clear that if it happens again, they will have to do their assignment without the manipulatives. If it happens again, take the manipulatives away for the day. Let them try it again the next time the manipulatives get pulled out with a clean slate, but repeat the process if necessary. Being firm about this at the beginning of the process will save headaches down the line.
At the end of the class, make sure you've scheduled time for the students to put everything away properly. You can enlist student helpers if necessary, but it is not your job to clean up after them. No matter how old the student is, they are old enough to put things back!
When Should We Use Manipulatives?
Manipulatives should not only be used during Centers or Stations. They should not be used sparingly. They should be available to students as often as humanly possible. Students will wean themselves off of the manipulatives when they start making the connections between the object, the symbol, and the mathematical idea both represent. Additionally, students will begin to realize there are faster, easier ways to solve the problem that do not require using the manipulatives once they become comfortable with the concepts. If you are familiar with the 8 Standards for Mathematical Practice, you'll recognize what I'm about to say next. Our students need to learn which tools are appropriate to use in each situation. One way to help them with this skill is to provide students with multiple types of manipulatives and allow them to choose which one to solve the problem. Sometimes their choices will surprise you and you'll learn a new way to approach not only the problem, but the manipulative as well. Sometimes they will choose a manipulative that is not the most efficient, but that's ok. They will learn.
[Anecdote Time: I had my 6th graders measure their heights so we could calculate the mean height for their period as well as the entire grade. I provided them with a bucket of different tools to use that included rulers, measuring tapes, meter sticks, and calculators. As I circulated around the room, I noticed one student lying down on the floor with another student moving a small object along the length of his body. While I thought it was incredibly strange, I let it go because they were working well together. As I collected the recorded heights, we had a great conversation about different units. One student was 5 feet 8 inches. Another student was 150 centimeters. Finally, the student from the floor told me proudly that he was 183 paperclips tall. I wrote it down, because it was technically correct. We got to extend our conversation about units to include nonstandard units, and the efficiency of standard vs. nonstandard units. The moral of the story: just because on the surface it seems wrong, doesn't always mean it is!]
Which Manipulative Do We Use?
Be careful about only using manipulatives for one concept and one concept only. Sometimes it seems as if a certain type of manipulative is only good for one type of skill, but that's rarely the case. For example, while Linking Cubes might seem like they are only good for geometrical models and exploring area and perimeter, they are also excellent for place value lessons, as well as exploring fractions and ratios. Algebra Tiles at first might seem like they should only be used for Algebraic purposes. If you compare the unit tiles to a Two-Color Counter, the only real difference is their shape. I'm in the process of creating a Google Doc with descriptions and activities for each of the more common (and some uncommon) math manipulatives to help in this particular area. If you have any suggestions for additions, please send them my way!
How do you use manipulatives in your classroom? If you don't use them, what are some reasons you don't? Share in the comments!
Why are Manipulatives Important?
Manipulatives help make math more concrete for students. Numbers always relate back to some amount of things, and allowing students to move those things around will help them understand what is happening when we apply different operations to them. Now, I know that most state tests do not allow for use of physical manipulatives, but students don't start the beginning of the year ready for those state tests. These are tools to help students build conceptual understanding of the math. Worst case scenario, the students use their scratch paper to draw representations of the manipulatives they used in the classroom throughout the year. Manipulatives also make math more engaging for students, and help them build their mathematical confidence. I know that in the beginning, it can seem like you're spending a lot of time trying to set expectations about how they should be used in class and trying to help students make connections between the math and the object, but in the long run the use of manipulatives in your classroom will cut down on the amount of remediation and re-teaching you have to do later on. This is because the students' understanding of the math will be more complete once they (and you!) get the hang of using the manipulatives in the classroom!
Classroom Management
Let's get the big scary stuff out of the way first. Children have a natural inclination towards playing. This is fine, as long as we are able to pull them back to the reality of the lesson and get them working. It's even better if we can trick them into thinking they're playing the whole time, while we're secretly teaching them. When you first give manipulatives to your students, allow them time to examine them and explore what they can do. You can structure this by including a question or two on their recording sheets asking them to describe the manipulative or write down things they observe.
Some students need a little help with organization, so providing a workspace for them is often very beneficial. This can be a piece of construction paper on their desk that their manipulatives need to stay on. You can extend this and require only the manipulatives that are part of their answer to be on the paper while the rest are on their desk. This will help you see what they are seeing, as well as keep things a little more orderly.
Some teachers have their students put both hands on top of their heads when it's time to listen to instructions. This will only work if you sell it to them, but even middle school students will buy into it if you believe in it. Having some sort of attention grabber and signal from the students is important, but make sure it fits with your personality and the personality of the class.
If the manipulatives end up in the air or on the floor in manners that are unacceptable, give a warning to the offender, and make it very clear that if it happens again, they will have to do their assignment without the manipulatives. If it happens again, take the manipulatives away for the day. Let them try it again the next time the manipulatives get pulled out with a clean slate, but repeat the process if necessary. Being firm about this at the beginning of the process will save headaches down the line.
At the end of the class, make sure you've scheduled time for the students to put everything away properly. You can enlist student helpers if necessary, but it is not your job to clean up after them. No matter how old the student is, they are old enough to put things back!
When Should We Use Manipulatives?
Manipulatives should not only be used during Centers or Stations. They should not be used sparingly. They should be available to students as often as humanly possible. Students will wean themselves off of the manipulatives when they start making the connections between the object, the symbol, and the mathematical idea both represent. Additionally, students will begin to realize there are faster, easier ways to solve the problem that do not require using the manipulatives once they become comfortable with the concepts. If you are familiar with the 8 Standards for Mathematical Practice, you'll recognize what I'm about to say next. Our students need to learn which tools are appropriate to use in each situation. One way to help them with this skill is to provide students with multiple types of manipulatives and allow them to choose which one to solve the problem. Sometimes their choices will surprise you and you'll learn a new way to approach not only the problem, but the manipulative as well. Sometimes they will choose a manipulative that is not the most efficient, but that's ok. They will learn.
[Anecdote Time: I had my 6th graders measure their heights so we could calculate the mean height for their period as well as the entire grade. I provided them with a bucket of different tools to use that included rulers, measuring tapes, meter sticks, and calculators. As I circulated around the room, I noticed one student lying down on the floor with another student moving a small object along the length of his body. While I thought it was incredibly strange, I let it go because they were working well together. As I collected the recorded heights, we had a great conversation about different units. One student was 5 feet 8 inches. Another student was 150 centimeters. Finally, the student from the floor told me proudly that he was 183 paperclips tall. I wrote it down, because it was technically correct. We got to extend our conversation about units to include nonstandard units, and the efficiency of standard vs. nonstandard units. The moral of the story: just because on the surface it seems wrong, doesn't always mean it is!]
Which Manipulative Do We Use?
Be careful about only using manipulatives for one concept and one concept only. Sometimes it seems as if a certain type of manipulative is only good for one type of skill, but that's rarely the case. For example, while Linking Cubes might seem like they are only good for geometrical models and exploring area and perimeter, they are also excellent for place value lessons, as well as exploring fractions and ratios. Algebra Tiles at first might seem like they should only be used for Algebraic purposes. If you compare the unit tiles to a Two-Color Counter, the only real difference is their shape. I'm in the process of creating a Google Doc with descriptions and activities for each of the more common (and some uncommon) math manipulatives to help in this particular area. If you have any suggestions for additions, please send them my way!
How do you use manipulatives in your classroom? If you don't use them, what are some reasons you don't? Share in the comments!
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