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!

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?