Math Tricks

How do we make sure our students are making the connections between concepts that are necessary for them to have good number sense and a solid understanding of the math they are doing?

My first year teaching I was worried that what I was teaching might go over my students' heads. The way I dealt with this was to use tricks and stories to help them understand things like solving one-step equations or dividing by fractions. I had learned some math concepts this way as a student, so I thought it made sense to teach it to my students. This made me feel like I was continuing in the grand tradition of math teachers, so I didn't really stop to examine what I was doing or why. It's just how it had been done. What I found was these tricks not only muddied the waters while they were learning with me, but very few of those tricks held up past my class. 

"When we ask children to follow a procedure that holds no meaning for them, they will conclude that math does not make sense. Go back to the mantra “math makes sense” and make sure all students believe it..." - Tina Cardone, "Nix the Tricks"

I was not setting them up for success down the road by telling them to "keep, flip, change" when dividing by fractions. I was also discounting my students' ability to understand what was really happening within math; I was insulting their intelligence and dumbing things down.

Let's examine putting the bigger number on top when subtracting as a simple example of why tricks are dangerous. Many of my students would recoil in horror if I put a problem on the board like 7 - 10. "You can't do that!" they would shriek at me, worried that the time/space continuum was about to collapse before their very eyes. At first, I couldn't understand their revulsion, until they explained to me that you must put the bigger number on top when you subtract. Since most state curriculums do not introduce negative numbers until 6th grade, I understand what their elementary teachers were trying to accomplish by teaching them this "rule." In all their experience, my students had started with a number and taken an equal or smaller number away from it. This "rule" worked for every situation they had encountered prior to meeting me. 

A more concrete way to teach subtraction is to give students manipulatives such as ten frames or unit cubes or base ten blocks. This way, they can start with one amount and take some amount away. If they need to regroup, they can break apart a ten or a hundred physically. Another method is to have students model subtraction on a number line. This method also allows students to explore negative numbers more easily and naturally. Once students understand what is going on when they subtract, moving to the standard algorithm becomes easier and makes more sense.

I came across Nix theTricks as a second year teacher, which helped me explore different methods for teaching math which were simpler and more straightforward. I have tried over the years to break my own bad habits of teaching math concepts in unnecessarily complicated ways, and each time I am successful, my students benefit greatly. Not only does it save time during the initial lessons, but when we need to use previously learned math in order to problem solve and learn something new, my students have a firmer foundation and a better understanding that allows them to make new connections.

What are some tricks you have stopped using in your lessons? Share in the comments!

Number Talk Discussion #10

186 ÷ 6

There are a number of ways to approach this problem. Before we look at them, let's define each term of the division problem. 

186 is the dividend, or the number to be divided by another number.

6 is the divisor, or the number you divide by.
The answer to the problem (31 in our case) is called the quotient.

Multiply Instead

"I know 6 times 30 is 180, plus one more 6 gives me 186; so my answer is 31."

Expanded Form (Chunk it Out)

Make a Tower

Halving and Halving

Which of these methods work better for this problem than others? Did you try any methods not listed here? Share in the comments below!

Friday Number Talk #10

Solve the following problem using two different methods.

186 ÷ 6

Share your methods in the comments below!

Number Talk Discussion #9

146 + 197

Round and Adjust

Take and Give

Start from the Left

Break one Addend Apart

Add Up

Did you try a different method? Are some methods more efficient for this problem than others? Share in the comments below!

Friday Number Talk #9

Solve the following problem in two ways.

146 + 197

Share your methods in the comments below!

Number Talk Discussion #8

43 + 9

There are a number of ways to approach this problem. Before we look at them, let's define each term of the addition problem. 

43 and 9 are both called addends.

The answer to the problem (52 in our case) is called the sum.

Round and Adjust

Take and Give

Start From the Left

Break One Addend Apart

Add Up

Did you try a different method? Are some methods more efficient for this problem than others? Share in the comments below!

Failure Should be an Option

I recently got my hands on an amazing article by Edward Burger about his experience with grading students in his college courses on their ability to fail.

If students are afraid of making mistakes, it means they are afraid of struggling, of trying something new, of being creative, of thinking in a different way. If grit is the key to success, we teachers are well positioned to nurture the quality.

The fear of failure is a learned one. As teachers, we are often so strapped for time that we don't think spending time on mistakes is a good use of what little time we have; we need to make sure the student can get the right answer so we can move on to the next topic. I think that mistakes allow people an amazing opportunity to ask themselves,  "Why didn't that work? Would it ever work?" Unfortunately, in education we tend to ignore those opportunities and instead focus on getting the correct answer. But then we bemoan our student's inability to think critically or persevere when confronted with something slightly more challenging than the last thing they did.

So what does the research tell us about how to allow students to be incorrect and still use our time efficiently? How do we build in time and opportunities for productive failure? The classroom environment we cultivate can have huge effects. Edward Burger provides his class with an end of course grade for how well each student failed. He asked his students to write a personal reflection essay about their experience with their own failure throughout the course and to give themselves a grade from 0 (did not make any mistakes) to 10 (made lots of mistakes and learned from them). He ended up using the students' own assessment of themselves for their final grade, which counted as 5% of their final grade. The students who received the highest grade were the students who admitted to having moments of failure and were able to reflect on those moments and learn from them. Other teachers have shared personal stories of their own failures and what they learned from those failures at the beginning of the year.

In her book, Rethinking Grading, Cathy Vatterott compares the learning process to being an actor in a play. An actor has many rehearsals (practice) and receives feedback on how to improve their performance during this time. The dress rehearsal (formative assessment) serves to show the actor how close they can come to getting it right. Finally, the actual performance seen by the public, serves as the only thing that "counts," or the thing that gets the grade. She also reminds us that "learning is not error-free - mistakes will be made. If we want to encourage students to view mistakes as a necessary step in learning, we need to remove the threat of grading while they are learning." Instead, she encourages teachers to provide "informative and nonjudgmental" feedback to students during the learning process.

Help create an environment that encourages productive struggle. 

• Make posters with quotes from famous inventors, athletes, and business people to decorate the classroom. Include quotes about failure and perseverance.
• Praise students in a way that helps them feel more comfortable making mistakes. Some suggestions include, "Wow, you really practiced that, and look how you've improved." "See, you studied more and your grade on this test is higher." "You tried different strategies and you figured out how to solve the problem." "You stuck to this and now you really understand it."
• Use feedback instead of grades during the learning process. 
• If your students ask you a question that you don't know the answer to, tell them, "I don't know, but we can find out!" Let them see what it looks like to not know an answer and what it looks like to discover it.

In past semesters, I have tried a few different ways to elicit incorrect responses from my students, borrowing heavily from great educators such as Dan Meyer and Carol Dweck. I have praised the effort and work my students have put into solving a problem rather than how smart they are. I have presented math problems, and rather than asking for the correct answer, I've asked them for answers they know will be too high or too low, and then asked other students to explain why that particular answer is reasonable. After reading this article, I am excited about creating a culture in my classroom that encourages mistakes and reflection.

What ways have you encouraged your students to be wrong in your classroom in order to help them learn? Share in the comments below!

Six Assumptions Teachers Make

We all make assumptions about things our students know and don't know. Sometimes these assumptions are correct, and present no problems. Often, however, those assumptions are incorrect and throw off our carefully planned lessons. Today's post is about some of those assumptions, and ways we can slow down and plan differently to help our students and ourselves succeed.

We assume that our students know...

1. How to behave in class.  No matter what age we are teaching, we tend to assume that our students know how to behave. When students fail to meet behavior expectations, more often than not we respond by sending them out of the room or assigning consequences. While consequences are important, we need to make sure our expectations for behavior are clear and consistently implemented in order to set our students up for success. Taking the time to reteach those expectations after any school breaks, or if things seem to be getting generally out of hand, will ultimately save time in the long run by minimizing disruptive behavior. Additionally, if the expectations you set at the beginning of the year do not seem to be working, you can always change them, as long as you make it clear to the students what changes have been made.
2. How to take notes.  Children are not born with the ability to take organized notes. This is a skill that is taught to them. No matter how old your students are, assume that they do not know how to take notes, and teach them that skill. I used interactive notebooks in my classroom, and would project exactly what and how the students needed to copy the notes into those notebooks. I taught 6th grade, so there was very little leeway given to my students about how they were to take notes. This worked very well for my classroom. If you teach older students, they might have a method that works especially well for them. If that is the case, allow those students to keep doing what they're doing. However, this will not be true for all of your students, so make sure you set up a method that works for you and teach that to your classes.
3. How to study.  Because students tend to take disorganized notes, they do not know how to use those notes for future studying. Using lessons to help the students practice using their notes to study is a great use of time. Some ways to set the lesson up could include letting students use their notes to play a game with a partner such as Battleship, Flash Card Flip, Flash Card Match, I Can..., or Task Card Pass. Additionally, you could set up a notes scavenger hunt by creating a crossword puzzle that uses the terms from notes and the definitions as clues. 
4. How to manage their time.  Many teachers, especially elementary teachers, think that by giving homework they are teaching students time management skills. This is another unfortunate assumption. What ends up happening is the parent's ability to manage their time wisely is tested instead. We can teach students time management skills in a better way by timing activities in our classroom and allowing students to see the timer we are using. Timers can be used for everything from bell ringers, to independent practice time, to bathroom use (if you are a teacher that takes the entire class to the restroom at various times). If students are reading a long passage or solving multiple problems, we can help them by writing on the board where they should be at different time intervals. For example, "In 5 minutes, you should have finished the first paragraph," or "In 6 minutes, you should have completed the first three problems."
5. How to read on grade level.  Depending on the type of school district you are teaching in, you might have anywhere from 5%-60% of your students reading well below grade level. No matter what subject you teach, we are all reading teachers. Every subject requires students to read, even if it is only the directions. By making sure you are supporting what the ELA teachers are doing in their classrooms, you are also supporting your students and yourself in your own classroom. This support can be as simple as rewriting a text to match the student's reading level, or if the majority of your students struggle, you can close read passages together.
6. How to solve basic math problems.  Students often enter their middle school years with a poor conceptual understanding of basic mathematical operations such as subtraction, division, and multiplication. This causes many other issues later on, as one might expect. By using manipulatives and illustrations to make the math more concrete, students begin to understand the operations better. While many of us were not taught using manipulatives, or maybe even how to use them in our classrooms, it is worth the time needed to explore them for ourselves and the time needed to allow our students to use them in class. It is a much better use of our time than spending weeks and weeks trying to remediate and reteach concepts later in the year.
7. How to think about the text or problem being presented to them.  Again, this is a skill that is learned as we grow up, which means that as teachers, it is our responsibility to teach it to our students. We can do this by coaching students to ask themselves questions as they are working on their independent practice. For math, those questions might look like: 1. What information do I know? 2. What am I being asked to find? 3. What operations can I use to solve this problem? For any class that requires reading comprehension, those questions might look like: 1. What is the author telling me here? 2. Are there any hard or important words? 3. What does the author want me to understand? [For literature: 4. How does the author play with language to add to meaning?]

What assumptions have you made that you later realized were incorrect? How did you go about correcting those assumptions? Share in the comments below!

Before the First Day of School

As the school year is winding down, now is the time for reflection and self-evaluation. What worked this year? What didn't? What changes would you like to make in your classroom for next year?

If you are about to enter the classroom for the first time, now is the time to start thinking about how you want to set up your room. Looking to other teachers is the best way to figure out what will work and what won't. Talk to veteran teachers about your plan and listen to their feedback. Sometimes our ideas don't actually pan out very well in practice, and this is one of those times where learning from someone else's mistakes is going to make your life much easier.

I've created a checklist to help you get organized and hopefully help reduce stress and feeling overwhelmed by what you need to do before the students walk in on that first day of school.

What are some things you would like to try next year?

What are some things you did this year that worked well for you?

Share in the comments below!

Number Talk Discussion #5

247 - 98

Round the Subtrahend to a Multiple of Ten and Adjust (Katie's method)

Decompose the Subtrahend (a variation of Tommy's methods)

Add Instead

Same Difference

Break Apart by Place

Did you use a different method to solve this problem? Are some of these methods more or less efficient for this type of problem than others? Share in the comments!

Taking Care of Yourself as a Teacher

When you are a teacher, it is easy to lose sight of your own needs. It took me until Spring Break of my third year in the classroom to realize I hadn't been taking care of myself my entire career as a teacher. If I took time off over the weekend to sleep or visit family or go hang out with friends, I felt guilty. I celebrated the first time I went into a store and did not leave with something for my classroom or students with a Facebook post and some delicious Mexican food. This was well into my second year of teaching, and I still remember it vividly.

If you are about to become a teacher, or have been a teacher for years, don't fear. This is not a rant about how hard teaching is or all the burdens we take on in that role. This is a post about the ways you can take care of yourself as a teacher. Or as a human person. Some of these things might seem like common sense, and if they are things you already do, Hooray! Keep doing them! My hope is that I will give you some new ways to take care of yourself during the school year, and every day outside of it as well.

Take Care of Your Health

• Eat healthy snacks throughout the day.
• Stay hydrated.
• Use the restroom when you need to! (Ask another adult to watch your class if you have to. Do the same for others if they ask you.)
• Take vitamins.
• Wash your hands, wipe down desks and doorknobs, and use hand sanitizer if soap isn't available. Do this religiously!
• Get enough sleep. Schedule time for this if you have to. Set alarms on your phone to remind you that it's time for sleep. Sleep is important!!

Take Care of Your Emotional Wellbeing

• Write (poetry, blog, diary entries, letters, etc.)
• Read good books (and not just books about teaching!)
• Go out with friends. Try to go out with friends who aren't teachers occasionally!
• Make something (cookies, cakes, food, bread, paintings, drawings, pottery, etc.)
• Watch TV and movies.
• Go for a walk/run/bike ride.
• Meditate
• Go to a counselor regularly (this job is hard, so let someone else help!)
• Talk to friends and family who aren't nearby
• Keep a journal at school and document the "moments of win," no matter how small they may seem.
• Make positive phone calls home and send good notes home with students.
• Set a work schedule and stick to it. If school lets out at 4, stay until 5 and work. After 5, if it's not finished, you'll get to it tomorrow. If you've scheduled 3 hours for work on Saturday, you only get those 3 hours. After that time is up, you have to go do something else.

Remember

• Never feel guilty for taking care of yourself first! You cannot be an effective teacher if you are sick or tired or stressed out all the time.
• Mistakes will be made. Learn from them and move on. Do not strive for perfection, strive for "better than yesterday."

What are some things you do to take care of yourself? Share them below in the comments!

Number Talk Discussion #4

34 - 9

There are a number of ways to approach this problem. Before we look at them, let's define each term of the subtraction problem. 

34 is the minuend. The minuend is the first number in a subtraction problem. The number from which another number is to be subtracted. 

9 is the subtrahend. The subtrahend is the number that is to be subtracted. The second number in a subtraction problem.

The answer to the problem (25 in our case) is called the difference.

Round the Subtrahend to a Multiple of Ten and Adjust

Decompose the Subtrahend

Add Instead

 

Same Difference

Break Apart by Place

Did you use a different method to solve this problem? Share in the comments!

Friday Number Talk #4

Solve the following problem using two different methods.

34 - 9

Share your methods in the comments below.

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:

1. identifying variables in the situation and selecting those that represent essential features,
2. formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,
3. analyzing and performing operations on these relationships to draw conclusions,
4. interpreting the results of the mathematics in terms of the original situation,
5. validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable,
6. 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.

Additionally, many teachers have not been properly trained on the new standards many of their states have adopted, or been given proper time to review and discuss the changes to the standards, and that leads to strange questions being asked, especially in math. For example:

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?

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 Number Talk #3

Without counting one by one, figure out how many boxes there are.

Share your method in the comments below.

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?

Friday Number Talk #2

Without counting one by one, figure out how many dots there are.

Share your method in the comments below.

A Discussion About Homework

Homework is a hot button issue for parents, students, teachers, and administration.

• 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?

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!