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!

Friday Number Talk #8

Solve the following problem in two ways.

43 + 9

Share your methods 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!

Number Talk Discussion #7

25 x 16

Break a Factor into Two or More Addends

Factor a Factor

Round a Factor and Adjust

Halving and Doubling

Connecting Arithmetic and Algebra

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!

Friday Number Talk #7

Solve the following problem in two ways.

25 x 16

Share your methods 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!

Number Talk Discussion #6

14 x 12

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

14 and 12 are both called factors.

The answer to the problem (168 in our case) is called the product.

Break a Factor into Two or More Addends

Factor a Factor

Round a Factor and Adjust

Halving and Doubling

Connecting Arithmetic and Algebra

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!

Friday Number Talk #6

Solve the following problem in two ways.

14 x 12

Share your methods 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!

Friday Number Talk #5

Solve the following problem in two different ways.

247 - 98

Share your methods in the comments below!

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!