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After my second year of teaching my state switched to using the Common Core State Standards.  While the standards weren't too different from the standards we were already using, how they wanted us to teach math switched drastically.  No more feeding students algorithms, they wanted students to discover, apply and connect.  I was totally on board with this switch, but the problem was ALL of our textbooks were now considered old. I suddenly had no resources for my students to use.  However, that summer before school started I went to various classes that taught about how we would now be teaching math.  They also addressed the issue of our lack of resources.  They taught us how we could actually use our old resources but we just needed to reverse the questions.  This idea was brilliant!  Let me give an example,  a math question may have said something like, "Find the volume of this rectangular prism with a height of 3 inches, a width of 2 inches and a length of 10 inches."  Instead, reverse the question, "Create a rectangular prism that has a volume of 60 cubic inches. Justify your answer."  So much more reasoning goes into the second question.  

I used this idea of reverse questioning and created an activity called "What's the Question?"  Essentially, I give students the answer to the question, and they have to come up with the question.  Many times there is more than one answer, but as long as students can justify their reasoning it works for me.  

I am giving away a sample page so you can get an idea of this activity, and use it with your students.  You will see deep-thinking increase in your classroom!

CLICK HERE for "What's the Question?" sample page

I've been thinking a lot lately about statistics.  I have no stats on my thought, but I was thinking that statistics has to be the most versatile major.   If you majored in statistics you could probably look for a job in virtually any company.  Statistics is so vital and so useful for companies.  Also, on that thought, I recently read an article how statistics would be better taught in the social studies classroom. While I don't 100% agree with this idea, the author makes a good point.  Social studies is statistics in action. 

However, I think a better argument would be that math educators need to do a better job of applying statistics to the real world.  I have not had the opportunity to be in math classes around the country, although I would love to, but my overall feel is that statistics is being taught on a very superficial level.  I feel this because that's how I was taught, until my AP statistics class as a senior in high school, but up until that point statistics was very superficial.  Yes, I could find the mean, median, and mode, I could make a histogram, and even a box-and-whisker plot, but I had no idea why I would want to do those things.  I had no idea how much information I could pull from data.  I had no idea how truly useful statistics can be. 

The Millennial Generation is the generation of entrepreneurs.  As business owners, using statistics correctly can make your business flourish.  Ignoring the statistics of your company can cause your business to fail.  We can give our students a strong advantage for their futures if we delve into statistics with them more effectively.  

I'm interested in other people's opinions on my thoughts.  Feel free to comment!
Do you feel like you are constantly teaching and re-teaching how to solve equations?  Try this process and watch your students' eyes light up with understanding.  

Teaching mathematics should take the form of concrete -> symbolic -> abstract.  If you just jump right into teaching abstractly you will not reach all of your students.  In this post I will review how you can take solving equations through these 3 steps.  I have used this process in my classroom, and it has proved to be very effective.

I will go through this process with the equation x - 3 = 10

Before going through the process emphasize the meaning of the equal sign.  Many students will think that the equal sign means "the answer is".  Teach that the equal signs means that both sides are the same.  Many teachers relate this to a scale, which is a great visual.  The scale will become unbalanced if you only add or subtract from one side of the equal sign.

To concretely solve this equation have students use Algebra tiles.  Tip:  Have students circle the terms separately, this will help them to not be confused with the signs.  Hopefully you have already talked about the additive inverse when teaching integers, if not, teach this property.  Tell students they can add or subtract anything from both sides until the variable is alone. 

Now you will move to drawing symbols for the tiles.  I often still let students use the tiles if they need it to guide them in their thinking. I will have them draw a symbol for each tile.  Many students start by actually drawing the blocks, but they soon change to just writing the "1" or "-1". 

Next you will move to abstract.  Instead of writing "1  1  1" students will write "+3".

One more tip:  ALL students should start at the concrete level.  Allow students to move through the progression of concrete, symbolic, abstract at their own pace.  Allowing students to take they time they need at each level will help students to develop a deep understanding of the mathematics.  

This post is also featured on the TpT Blog

1)  Help students make connections between different math topics, especially Algebra and Geometry.  Many people think of mathematics as discrete topics, this is detrimental to students' learning.  As you study mathematics you will learn that math is intricately connected.  Helping students make connections will help them make sense of math and retain the material.

2)  Beware of giving your students algorithms. Students may be able to memorize a few lists of step-by-step algorithms that you give them, but do you expect them to be able to remember ALL the steps for every algorithm?  What about the students that have difficulty memorizing?  I'm not against algorithms, I'm just against giving step-by-step algorithms to your students.  Instead, give them a problem and let them figure it out, then have a discussion with them about what they noticed in their process.  Guide them to discover the algorithm.  Doing this will help them make sense of the mathematics, and internalize the algorithm.

3)  Get writing.  Have your students explain their thinking as much as possible.  Teach them to use mathematical vocabulary as they explain.  Students will often resist writing in math class at first, but be consistent and show good and bad examples so they know what you expect of them.  If you continually require written explanations of their math then your students will internalize the mathematics better.

If you are looking for some resources to get your students writing check out these writing prompts.
Click Here for 7th Grade Writing Prompts
Click Here for 8th Grade Writing Prompts

      About two years into teaching middle school math, I realized a HUGE mistake I had been making.  I wasn't teaching something that was very important.  I never thought to teach this topic, it wasn't explicitly written in the curriculum.  However, I noticed this was a problem by the questions I started receiving from students while teaching. I realized I needed to take a day or two and explicitly teach this.  To me, it was just something I knew and picked up, but I realized not everyone picks it up the same way.  This topic is parentheses notation.  Yes, I explained that parentheses also meant multiplication, but that's about as far as I went.  

     Parentheses notation can actually be very complex, and many math teachers likely don't realize the confusion this can cause for students.  For example, comparing the two equations 6(-2) and (6)-2.  SO many similarities between the two expressions, yet so different in meaning.  Or are they different?  What exactly am I trying to say in the second expression?  Six take away two, or the product of 6 and -2, and what does it depend on?  This can be SO CONFUSING for some students.  Other students will just know, and they may not even know how they know, but they will just get it, others need parentheses notation taught explicitly. Take the time to teach parentheses notation, you do not need to spend a whole unit on it, but at least spend a day.  This will help students in the long run.  I made a "Preventing Parentheses Pitfalls" resource to teach this very subject.  I have decided to make it FREE to all fellow math teachers in hopes that they will take the time to teach this topic.  Click below to download yours now.


          As a math teacher, I can't even tell you how many times a student would excuse their poor math work with the comment, "Well, I'm just not a math person."  What was even more horrifying, is when a PARENT would excuse the poor math work of the student with the comment, "Well, I'm not a math person, so he/she is not a math person."  There does not exist two categories of math people or not math people.  However, I do believe that there exists two categories of people who know how to learn math and people who do not know how to learn math.  The great thing is that these categories are flexible and you can easily teach your students to belong to the "I know how to learn math" category. Here are 7 steps to help your students be successful in the math classroom. 

1    1)    Daily  engagement
Stress the difference between engagement and participation.  Participating students may simply be copying notes. Engaging students may be copying notes and trying to internalize the notes by making connections.  Engagement encourages the use of higher-order thinking skills. In order for students to engage daily, your classroom instruction needs to promote critical thinking skills.
     2)    Learn from mistakes
Encourage students to never erase mistakes.  Instead have them leave their mistakes, and with a different color they can mark and explain their mistakes.  Continually model this to students by marking your mistakes on the board.  A safe environment is required for students to feel safe to do this step.  Celebrate mistakes as a step in learning.
     3)   Ask  critical questions
An example of a non-critical question is, “What’s the next step?”  An example of a critical question is, “How do ratios connect with the circumference of a circle?”  Make a poster of words that help create critical questions.  You could teach them Bloom’s taxonomy, and classify different questions for each level.  Consistently point out and praise critical questions in the classroom.
     4)   Show all your thinking
Teach students different ways to show their thinking.  This can include in writing, with models, diagrams, equations, expressions, etc... Showing calculations depends on the level of the student.  Teach students to write in complete sentences.  Students should label their models and diagrams.  Do not accept low quality with this step.  Consistently push the students to do more and more.  Have them redo the assignment over and over until they are showing quality work. 
     5)   Don’t cut corners
Students often just want to “be done” with the problem.  To help students to not cut corners, assign fewer problems, but require quality.  Cutting corners causes students to make mistakes and not critically think through the problem.
     6)   Make connections
When students make connections they will retain the information more easily.  Many times connections are not obvious and you will need to guide them to discover different connections.  Connections between algebra and geometry are critical to understanding higher-level mathematics.  Consistently push them to find connections.
     7)     Be humble
Humility is essential for students to learn mathematics.  The students that think they are “bright” are often those students who learn very quickly, mostly because they can memorize.  These students often don’t think they need to explain their thinking, because they already have the correct answer.  Don’t let these students cut corners.  Push these students to ask higher-order thinking skills.  The students who struggle often don’t want you to know that they will struggle, so they will erase mistakes and try to cover up their weaknesses.  Having a positive environment that values mistakes will help these students.

When the new core was implemented in my state I was heading into my third year teaching.  I attended an intense 80 hour course on the new material as well as the new strategies to teach math.  I fell in love with the new strategies that I learned and I was excited to implement them in my classroom.  However, I was caught off guard when I was told that the 8 Mathematical Practices would be tested. I had never actually explicitly taught mathematical practices, but I knew now was the time to start.  Through my years of teaching, I have done a few things in my classroom that have greatly impacted their learning of these practices.  Technically, these practices are supposed to be taught since early elementary grades, but even as a middle school math teacher, I always took the time to explicitly teach them.

Here are three ideas that have worked in my classroom.

Assign a Reading Assignment
That's right, I printed out the mathematical practices and their explanations and I assigned my students to read them.  I had them mark up the text, as though they might do in their Language Arts class.  I had them highlight the text, annotate the text, and write questions about the text.  I had them collaborate in small groups about the text and then we had a large group discussion.  Taking the time to do this, truly made the world of difference.

Practice the Mathematical Practices
An excellent time to explicitly teach these skills is the first week or two of school.  I used logic problems to practice these skills.  For example, I would give a logic problem to the students, I often did this in small groups, and have them work on it together.  Then I would have the small groups present their "viable argument"to the class.  The students would then focus on "critiquing their reasoning."  The purpose of the class was not the answer to the logic problem, rather teaching the mathematical practice of, "Construct viable arguments and critique the reasoning of others."  This is just one example, but can easily be applied to other mathematical practices.  

Post and Refer
I made posters for the mathematical practices and hung them at the front of my room.  I kept them there then entire year.  I included them in my teaching on a daily basis.  I would tell the students what skill we were practicing along with the new material.  I would also have my students tell me what skill they were practicing, and have them write about what mathematical practice skill they were practicing on the assignment.  The key for this to be successful is to refer to them and talk about them on a daily basis.  Let them become part of your vocabulary and the students' vocabulary.
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