2017 - Make Sense of Math
3 keys to become a better math teacher


Teaching MATH 
Teaching math is a continuous challenge, it's like reaching a peak only to see another higher peak. There is always more to learn and ways to improve. Here are three tips to become a better math teacher.

Math Teaching Tip 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.

Math Teaching Tip 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.

Math Teaching Tip 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.
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3 keys to become a better math teacher


    

My Teaching Mistake
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 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 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. 

It Can Be So Confusing
     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 clearly. 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.  Grab it in my FREE RESOURCE LIBRARY




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Teach your students to master mathematics

Helping Your Students Master Mathematics      
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) 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 way 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 think that they don't need to explain their thinking, because they already have the correct answer. Don't let 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 struggle, so they will erase mistakes and try to cover up their weaknesses.  Having a positive environment that values mistakes will help these students.  Students who struggle may not want you to know that they struggle, so they will erase mistakes and try to cover up their weaknesses.  Having a positive environment that values mistakes will help these students.

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Teach your students to master mathematics




Implementing Mathematical Practices in Your Classroom

Mathematical Thinking
One of the purposes of math teachers is to help your math students develop mathematical thinking. This is achieved through implementing the mathematical practices.
  • Make sense of problems and persevere in solving them
  • Reason abstractly and quantitatively
  • Construct viable arguments and critique the reasoning of others
  • Model with mathematics
  • Use appropriate tools strategically
  • Attend to precision
  • Look for and make use of structure
  • Look for and express reasoning in repeated reasoning
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 to teach the mathematical practices. 

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.

If you need some mathematical practices posters then you are luck.  I created these mathematical practices posters that I absolutely love! 

Click on the image below to check out the mathematical practice posters. 


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Implementing Mathematical Practices in Your Classroom








Learn all about how to incorporate higher order thinking questions into you classroom to promote higher level thinking. A simple strategy to easily incoroporate these types of questions into your lesson plans. 

The Need For Better Math Problems

Many math teachers know they need to step up the quality of math problems in their classroom, but are not sure how to do it.  I was taught a great strategy to switch up my type of questioning and would like to share it with my fellow math teachers.   

The Problem

Teachers are often given math textbooks to teach from, and sadly, open-ended questions are not apart of the majority of math textbooks. However, changing common math problems into open-ended questions is very doable. Here is an idea that I have been taught that has encouraged higher level thinking in my math class.

The Solution

Take a common math problem and flip the question and the answer. Students can create questions with the given answer. Take the problem one step further by having them justify their answer either through writing or modeling (or both.)

Example 1

A common math question might be the following:  "Find the volume of a box with height 3 inches, width 5 inches and length 10 inches."  If you are familiar with Bloom's Taxonomy, a taxonomy of learning,  I would suggest this type of question falls into one of the lower levels. Students simply recall the algorithm and calculate the answer. 

While I do believe these types of questions have their place in a math classroom, they should be in the minority.  What if you instead changed the question to this, "Create a box that has a volume of 150 cubic inches." This question is now a critical thinking question that requires higher-order thinking skills. 

Example 2

Instead of, "Solve the following equation 3x + 2 = 11," flip around the question and say, "Create a two-step equation where the variable equals 3.  Write your equation in two different ways."

Example 3

"Find the mean of 12, 15, 18, 20 and 30."  Flip around the question and answer and ask, "What five numbers have a mean of 19 and a range of 18?  Justify your reasoning."  

Example 4

"Simplify the expressions 2(x + 1) + 4."   Change this problem to "Write three expressions that simplify to 2x + 6.  Prove that your expressions are all equivalent."

The Results

Flipping questions does take time, and sometimes it is "easier" to assign the basic problems.  Remember, however, that having students problem-solve and make sense of math will require less review and better retention.  I can assure you that doing so will definitely be worth your time.  

Also, one benefit of doing these type of questions, is there is often more than one correct answer. This can lead to great class discussions. 

Activities For You

If you want a head start on using reverse questioning in your classroom I have some great no-prep activities ready for you. Check out the following activities. 
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How to create higher-order thinking questions for your math classroom

 




How to step up the rigor in middle school math


Teaching Middle School Math Like a Pro
Starting my third year of teaching I realized it was time to step up my game as a teacher.  I finally felt that I had the basics down enough to run my class smoothly, now I wanted to become a better teacher as far as teaching my content.  That summer before my third year, I took a week long middle-school math teaching course.  My eyes were opened to everything that I was doing wrong.  However, I didn't feel upset, I felt excited.  I felt so excited that I finally knew how to become a better teacher.  My BIGGEST mistake was that I was teaching to my low students.  I thought in my head, well if my low students can understand then that will mean that all my students will understand.  This thinking is not necessarily wrong, it's just not very effective.  You see, for my low students to understand the content I had to significantly decrease the rigor of my classroom teaching, and the rest of the students suffered and were not reaching their potential.  I needed to step up the rigor of my classroom.  Here are 4 ways that I stepped up the rigor in my math classroom.

1 - Believe in your Students
I didn't believe in my lower students.  I didn't believe that they would understand if I didn't teach to their level.  Wow, was I wrong!  Our school director consistently taught us as teachers that if we believe in our students then they will believe in themselves.  I finally took this teaching to heart that year.  Instead of seeing them as "low" students, I started seeing them as students with lots of potential who could achieve and understand this content.  This mental shift was not a one moment occurrence.  There were various times throughout the year where I questioned whether or not I should simplify the content.  I had to consistently remind myself that they could achieve. 

2 - Target your High Middle-Achievers
Instead of targeting my low achievers, I started targeting my high middle-achievers.  I gave extensions to my high achievers so they were still being challenged, but I didn't feel comfortable making my high achievers my target audience.  I taught at a higher level, and do you know what happened?  The lower achievers rose to the challenge.

3 - Vocabulary
Using correct vocabulary may seem as though it is not as important in a math classroom, I highly disagree.  From day 1, I started pushing correct vocabulary.  I banned the words "it" and "thing" or any other vague word.  I forced myself to use correct vocabulary and I forced my students to use correct vocabulary. I did not simplify my math language into non-math terms. If needed, I spoke "above" their math level, because I wanted them to learn math correctly.   My students actually enjoyed this change, I believe they felt more intelligent as they started speaking more intelligently. 

4 - Open-Ended Tasks
This was a game changer in my classroom.  If you do not know what open-ended tasks are let me briefly describe them.  Next week I will write about how to write and incorporate open-ended tasks more specifically into your classroom.  Open-ended tasks are story problems that contain all/most of the following criteria:  have more than one correct answer, students may need to make some assumptions to finish the problem, there is more than one way to solve the problem, students prove their work, real-world application.   
I believe that there is a place for fluency practice in your classroom.  There is still a need to practice procedures, but DO NOT ignore the need to do extreme reasoning and problem-solving.  This practice is where students truly will make sense of math and stretch their brains.  

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How to step up the rigor in middle school math


How to teach for retention in your middle school math classroom


Middle School Math Teaching Tips
I was two years into my teaching when the new core curriculum was rolled out.  I was excited about the rigor of the new curriculum, but stressed out about how I was going to teach all of it.  I had mapped out the year various different ways to try and fit all the curriculum in the time allowed, including a few weeks for review at the beginning of the year.  I just couldn't make it work.  I talked to my director about my concern, and she very wisely asked me, "Why do you have to review at the beginning of the year?"  Her question caused me to do some deep reflection. Why do I have to review at the beginning of the year?  Are we as teachers teaching so poorly that the students need so much review.  She suggested that I review small concepts as they come up during the teaching of other topics, but that I didn't need to devote so much time to review as I had originally planned.  Also, she helped me to realize that I needed to teach smarter.  I read and implemented lots of strategies and I learned some key strategies to teach for retention.

1 - Connect the Concepts
Connect both across topic and connect linearly.  For example, connect Algebra with Geometry.  Connect Algebra with Arithmetic. The more we as teachers help students to see the connections they will make sense of the math.  As students make sense of the math their retention increases.  Also, as you connect the mathematics there will be a natural constant review of concepts.

2 - Teach Deeper
Instead of assigning 20 algebraic equations to solve, assign 4 in-depth questions that really cause the students to critically think and analyze.  If you are familiar with Bloom's Taxonomy aim for the higher levels. I like this diagram as the words help me to create questions that use higher-order thinking skills.

3 - Believe in Your Students
Step up the rigor in your classroom and BELIEVE that they can achieve.  If you believe in your students then they will believe in themselves.  If you see success in your students, they see success in themselves, as you act like they can succeed then they act like they can succeed.

Implementing the above strategies will help your students retain mathematics. This may feel scary at first implementing these strategies, but I have seen success in my own classroom as I connected the concepts, taught deeper and believed in my students, and I know that your students can succeed too.

Below are some links to tasks that implement higher-order thinking skills and connect concepts.  Perfect to help your students retain mathematics.

Math 7 Task-Based Assessment Prisms Taxes and Inequalities


Math 7 Task-Based Assessment Rational Numbers Tables Graph

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How to teach for retention in your middle school math classroom


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