I savored the last year of teaching high school math and retired in May 2024. It has been bittersweet because I LOVE teaching. I have compiled 37 of my favorite teaching ideas. Many of the ideas are not necessarily for just math. Please enjoy and as always let me know if you have any questions or clarifications.

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I read Peter Liljedahl's book Building Thinking Classrooms in Mathematics this past summer and it has been transformational in my teaching. Getting all my students to go to a vertical surface in groups of three has been MAGICAL!!!! I absolutely love the idea of students thinking more on their own than myself blabbing on about the material. Brilliant! It is a must read that is appropriately called the #thinkingclassroom. Let me give you major things that I changed in my classroom this year.

1. Select RANDOM Groups of Three

The first thing is the randomness of your groups. This is important because groups get old fast. If a student has the opportunity to work in a new group of students they are more apt to participate and give and take instruction with their peers. I change groups every time I have students go to the vertical surfaces which is about 3 times a week. I hand a deck of cards to a student and they are in charge of shuffling the cards and then passing them out. See the video below. Now before you hand the student the deck of cards you have to make sure there are the right amount of cards for the students. I actually have 5 sets of cards, one for each class. Then I don't have to mess with it too much because of absences. I then have different locations that the Aces would go to, and the two's would go to and so on. (see the image on point 2) The deck of cards help the students know that it is not you that is making the groups. After they get into the groups with all their stuff, this is the seat and location for the rest of the day. Also, groups of three is perfect. Two is too little and four is too much, three is just perfect.

2. Have all Groups go to a Vertical Surface

This is tricky because you will need up to 8-10 spots in the room where students can do problems. I use the wall which have white board paint on it. I also use my windows which create a perfect surface to write on. Sometimes teachers buy a 4 foot by 8 foot white board panel at Home Depot and cut it up and put it on the walls. There is such a thing as plastic whiteboard sheets that cling or stick to the wall that work as well. Since the students are switching groups locations so much I put a small group location map on the wall for students to navigate every time we change groups.

3. Give Challenging Problems.

In previous years, I usually would build up to problems that were more difficult with students. However, I ended up not doing the most difficult ones because of time. So start with the challenging ones first and you set a standard right away with vertical surface problems that will be difficult but reachable. If you start with a challenging question, the process usually flushes out other easier concepts that are needed in the lesson. Also, it is good for students to be stuck and figure out how to proceed. This is what Liljedahl calls PROBLEM SOLVING! What do you do when you don't know what to do. So many times we take the challenging part out of their work and we as teachers do it for them. Let them be challenged and think on their own. Lastly, try to give open ended questions. For instance, create an equation of a line that has an x-intercept of (0,5) and only passes through quadrants I, II, and IV instead of a question like find the equation of a line through these certain two points.

4. How to Deal with Questions?

I think you better read Peter Liljedahl's book for this question. However, the one thing I have learned so far this year is that I try to redirect questions back at their teammates (their other group members). This gives power back to the group. I have found that I know when students are just looking for confirmation or when they really need to know how to proceed. When a student asks me if something is correct, I don't directly answer that. I tell them to check with their teammates or to look at what other groups around them have done. This is a good way to get dialog happening because many times other groups will have a variety of answers so students have to decide how to proceed. Liljedahl has a list of excellent replies to questions. See below. I refresh myself with this list all the time. My favorite is how did you do that?

5. Only one Whiteboard Marker per Group and Change 'Writers'

Only allow one pen per group. This forces collaboration. Also, make sure you call out for students to change who is writing the problem down on the board. Sometimes groups change on their own, but more often I've found that you need to call out the change. Sometimes I have a timer on my phone and it rings every 1-2 minutes for a change of writer. This time amount always depends on the problem. The non-writers are encouraged to bring their spiral (blank paper) up to the vertical surface to write notes with.

Benefits to Vertical Problems

Math talk is huge...even 1st period.

Movement in the classroom.

Students are willing to take math risks with new group members

Mistakes are easily changed via dry erase.

Students love to write on the wall

Class collaboration because groups can all see how all the other groups are doing the problem.

Did I say math talk? Huge!!

Students like to take pictures of their work for future reference

Challenges to Vertical Problems

Some students are not collaborative and thus do not help their teammates.

Off task groups

Students don't always write down the problems/solution. Thus they don't have notes from their work on the vertical surface.

Zoom Fatigue? Students still need to move. Maybe more than ever we need to encourage students to get up and move around during their online learning. A little break in the middle of your lesson or video or slide presentation would help. I'm planning on giving one brain break per class period.

Here are some ideas of how to use a brain break in remote learning.

In the midst of your lesson via Google Meet or Zoom time, put in a brain break slide. I have included 9 different of my favorite Brain Breaks in slide form. Do you present with SmartBoard or PowerPoint or Google Slides? Each of the following files have 9 Brain Breaks with video explanations. (Link to SmartBoard, PowerPoint or Google Slides Click Here)

In the middle of a video you have put together for your students, pause and ask your students to stand and walk through a brain break with you.

Do you have a Learning Management System (LMS)? Put a module in for a Brain Break.

Zoom Time: I have included a few partner brain breaks below. Explain the Brain Break (or show the video) Then separate your students into breakout rooms of 3-4 students each and give them 2 minutes to do the Brain Break.

Here are 9 of my favorites

These are partner brain breaks. I would suggest using breakout rooms in Zoom.

I bought a unicycle in June 2020. The Covid lock down encouraged me to do this. Then a couple of weeks later I bought a different size unicycle because each has it's own unique speed and handling abilities. I have spent the last month and a half learning how to ride them. It has been an amazing journey as I think of all the parallels to teaching high school math. Here are my big takeaways.

1. Why am I doing this? As a unicyclist...
The biggest question I get when people ask me about unicycling is "why?" My answer is that I have always loved cycling and I have always been intrigued about unicycling. Why not start now, during the pandemic? Interestingly, I have a lot of people communicating with me when I ride my unicycle. Things like, "Great work" or "That looks tough" but also I get "Where is the other wheel?" or something with the word "circus" in it. I'm not doing it for the attention I'm getting, however, I find that it is certainly is a conversation piece. As a teacher....
Students are not given a choice whether or not to take math. We as teachers need to build the case FOR mathematics. We need to inform students of some of the good reasons why they should have a desire to do mathematics. We should allow the question of "Why am I doing this?". I'm gonna try this...
Take a whole day at the beginning of the year and ask students to consider the WHY behind taking their math class.

I bought one Unicycle and then another a couple of weeks later. The different size changes the speed and handling My two unicycles

2. I am going to fall down. As a unicyclist...
I fell hard today when I was riding. Both of my elbows have scrapes on them. UGH! All it took was a loss of concentration for a second and bam, I was on the ground. It happened when I was on an asphalt path and I had just passed a person who was walking and had said, "Good job." Embarrassing...yes. But the hardest part was getting back on the unicycle. I did and ended up riding 6 miles today. Most falls are not all the way to the ground. I usually fall off right on my feet. As a teacher....
Anything worthwhile involves struggle. Productive Struggle. We will all fall down at points in our journey. This step is overlooked in math classes. Teachers and students like to AVOID STRUGGLE. We try to do everything we can to help students miss the difficult stuff. We must do better. I'm gonna try this...
Embrace struggle as a teacher. I'd like to try to have a student give the class (and me) a math question that we do not know answer to ahead of time. Then I can model the steps for trying to solve the problem.

My first day on the unicycle

3. Going straight is easy; trying to turn is difficult.

As a unicyclist...
I can go straight fairly easily now. As a matter of fact, if I'm on a flat surface and do not have to turn, I'm golden. I learned in a parking lot and found I could go whatever direction I wanted but usually went straight. However, when I have to turn on command, I'm in trouble. Bumps are another problem. If there is a bump in the road, or a small incline, it is much harder to ride. So now on my unicycle rides I have been making small challenges for myself. For instance, instead of going straight, I will incorporate a 90 degree turn or ride up a curb or even through the grass. As a teacher....
We have to challenge our students to try hard things. Things that are difficult are memorable, and things that are easy we tend to forget. We need to work with students to set and achieve small goals that would be difficult but not impossible to accomplish. I'm gonna try this...
I'd like to give my students a choice as to which option of difficult problems they want to tackle.

After two weeks of practice

4. Videos helped, but mostly I needed to try on my own. As a unicyclist...
I looked at a beginner unicycle video before I started. I was OVERWHELMED! So I just had to get out there and try riding. After my initial outing, I went back to my beginner YouTube video and gleaned some new information. This has been my pattern, even now. As a teacher....
I talk too much. The more I talk, the less my students get to practice problem solving on their own. The more I talk, the less I can give feedback to individual students. I'm gonna try this...
Give a SHORT INTRO VIDEO then have students TRY A PROBLEM ON YOUR OWN (with automated feedback)... then GIVE MORE VIDEOS. Don't overwhelm my students with a big, long video explaining everything. This is not going to stick and most of your students will be bored out of their minds.

After seven weeks of practice

I'm still learning to ride a unicycle. I thought when I was just beginning that I would arrive at some point and just be good at it. I'm realizing that unicycling/learning is a journey that the destination keeps changing as you get better. This is the fun of learning. We will never really arrive. I didn't magically learn how to ride a unicycle just like we don't magically 'get' math. The process of learning means that we will be falling down a lot. I'm ready for this challenge because I know why I'm doing this. Learning is fun and rewarding. And the next time I'm trying to do something other than go straight ahead, I will try to do something that is more difficult than I'm comfortable with.

James is a senior here at Naperville Central High School that I have known from when he as a soccer player on the freshman team that I coached. I recently had the opportunity to shadow James to see what it feels like to be a student at this school. He graciously allowed me to follow him around for an entire day of classes, even lunch and P.E., and I had a blast. What a great day!

The question that students asked me over and over all day was WHY? Why are you shadowing a student? My quick answer was, "I want to know what a high school student goes through every day." It was an eye-opening experience and I’m so grateful to our administration for encouraging me to do this.

My Big Takeaways
1. I sat in my seat for five 50-minute periods without moving. STUDENTS NEED TO MOVE IN CLASS.

2. I went with a group of friends to Dunkin' Donuts for lunch. We all brought our lunches in and some of the guys ordered donuts for dessert. I had never before realized the positives of going out to lunch. The change of pace that occurs from taking a break in the middle of the day was amazing. CHOICE and FREEDOM ARE LIFE GIVING.

3. I met Harold in Anatomy and Physiology class. We became very close in a short amount of time. You see, we were dissecting Harold the rabbit. That was a complete thrill. LEARNING BY DOING was one of my favorite things of the day.

4. I got called out for having the wrong shirt in PE class. Thankfully my dad did not receive a call from school. KNOW THE RULES.

5. Students are basically working from 7:45 to 3:15 at school—a 7.5 hour workday. And that doesn’t include the hours of homework. It is exhausting. WE TEACHERS GIVE TOO MUCH HOMEWORK. I'm glad I didn't have to do all the homework James had to do that night. I vow to lower my homework demands.

6. James was an amazing host. He answered so many of my questions. In every class there were respectful interactions. The students at Naperville Central High School are terrific. WE HAVE GREAT PEOPLE AT OUR SCHOOL

7. By “walking a mile in someone else’s shoes” I feel more in tune with my students' needs. Every teacher should shadow a student for a day! WE NEED MORE COMPASSION

EVERY ADMINISTRATOR, COUNSELOR, PARENT, and TEACHER should Shadow a Student for a day. Here is the hashtag I used on twitter #shadowastudent Click here to check out the tweets that day...

This is my third time shadowing a student. Here are links to the other blog posts about shadowing a student. The first was in 2012 when I shadowed Neal a freshman. The second time was in in 2014 when I shadowed Kyle a junior.

A student stayed after school this past spring to talk with me. I figured it was something about a grade or some help he needed. He actually wanted to talk with me about the math class SCRIPT. Here is what he said about math classes in general... ...everyday we had a note sheet, some set examples, a target written out with what we were learning, and then there were blanks so you would wait for the teacher to put it up on the board so you could then fill in the blanks. Then you just followed all the examples. It is like a script. I guess it was really efficient in teaching you exactly what you needed to know, but there was no thinking in it. The teacher wants you to follow exactly how they tell you...you can't deviate at all...I couldn't do it another way or skip this step.

As a math teacher, you know the script: review homework, present any tools/formulas you need for the lesson, then show how to do problems step by step using those tools/formulas and then give alone/group time to practice all of that. I know the routine very well because I learned math using this script when I was in school K-12. Once I become a math teacher, I modeled this script for many years. And it works! The script is efficient and mistakes are minimal. If the students do the work, they will be prepared for the test. I as the teacher gave the most efficient method to the students to solve a problem and then they practice. Many students like the script because it works for them. Students know what is expected and respond to it. If students practice enough, they can reproduce the work on tests/exams. If you look at the last few things that my student said: follow exactly how they tell you, don't deviate, don't do it another way. This is worth looking at because I don't always follow the script in my class anymore.

My student went on to say a few things about my class in particular.

...when I first took your class, and you suddenly threw us into this place where we have to do it and think ourselves, I was like...crap I'm not going to do good in this class.

Whoa. As you can see, it was not pretty in my class the first part of the year. I often have many students that are completely frozen with fear in my class because they have never had to do a problem without first being confident on how to solve it. I'm working on how to calm this fear, but in reality, this is a mathematical mindset that is important for all students to confront. I often ask my students to look and engage with a problem BEFORE they are given any kind of formulas. Here is the reason I do this...“Trying to solve a problem before being taught the solution leads to better learning, even when errors are made in the attempt.” ― Peter C. Brown, Make It Stick Furthermore, students tend to be more engaged with the problem and own it after they have tried it themselves. So when you give the problems to the students (in varied ways) before you have shown them how to solve it, they remember it better and they are more engaged and empowered. This is typically how I run my class.

And my student says a little more about my class.

...I got the hang of it, and then I started to realize ...Oh, I'm doing this automatically in other classes now.

What? What did he get the hang of? I explored this a little more with him and he said this...

...for example when I first started physics it was just really hard because I wasn't able to think freely. I wasn't able to think abstract or imagine things. Now I can think on my own and decide which way is the right approach.

He is evaluating his own work and practice. He is actually empowered by thinking on his own instead of just following a script.

I have been thinking about this all summer long. There are things in this conversation that are really important to note. One, he felt awful at the beginning of my class. This is important for me to help smooth out. Second, he felt liberated by going outside the script. And thirdly, he uses this technique in other classes. Since there are pros to the SCRIPT and there are pros to being off the SCRIPT, it is important to remember that students, parents, and other teachers are not always on board when you deviate from the traditional math class SCRIPT.

Benefits of the math class SCRIPT
Efficient
Routine
Do the work, and you are prepared for the test
Mistakes are minimal
All the expectations are given in a concise manner
Many students like the script because it works for them
All are on the same pace

Benefits of being outside the math class SCRIPT
Liberating
Empowering
Lack of direction creates a mathematical need
There are MANY ways to solve a problem.
Making mistakes and learning from them is OK and needed
Many students love being in charge of their own learning

I would say that I am using the SCRIPT less and less as time goes along. I'm forever tweaking my teaching to get better.

I encourage you to try a different approach. Make sure you tell your students what you are doing and why. Then give it a go. Give the students a problem before you have explained all that is needed to solve it. Give them time to work alone and then with others to solve it. ENGAGE. Lastly, discuss what math is needed to solve the problem and work through how students can get that information.

This game is the reverse of a Number Talk. In a number talk the teacher gives a problem and students find different ways to come up with the answer. With TRUTH or SNARE, the teacher gives an answer and the students create potential equivalent possibilities (truths) or non-equivalent possibilities (snares). The rest of the class has to find ways to determine if the possibilities are a truth or snare.

Let's give an example

Teacher: Let's play TRUTH or SNARE with the theme being exponents and order of operations. Remember that SNARES are NOT-EQUIVALENT but are really close to being equivalent and very tricky. Create at least 2 TRUTHS and 2 SNARES for the following answer: 16 (Teachers resist the urge to give examples. Don't do it. Instead, encourage the students to create a problem that has an answer of 16. Remind the students that it won't matter if their problem is equivalent or non-equivalent...they will not be wrong)

Students: Work in groups of 2-4 to accomplish the task. (Teachers make sure each group has at least one example of either kind)

Teacher: Pick groups at random to present their possibility. Students will not be telling the rest of the class if it is a truth or snare yet. Have the groups put their problem on the board or have the group give them to the teacher to put on the board.

Teacher: Now take some time in your groups to decide which ones are TRUTHS or which ones are SNARES. (Allow enough time for all groups to get through at least half of the examples given)

Students: Work together in your group or with your partner to determine if each problem is a TRUTH or a SNARE. Be ready to defend your work.

Class Discussion: Take a vote for each example. Thumbs Up (Truth), Thumbs Down (Snare), and Thumbs Middle (Unsure). (see my voting thumb pictures below) Have students walk through the reasoning and or the teacher walk through the reasoning for each example. Be open about the fact that this activity is meant to push your understanding and that you might make mistakes on what is a TRUTH or SNARE. These mistakes are helpful for us to recognize equivalencies in the future.

See the example below that we did in class.

Here are a few examples of what the students made.

Group B,C and D are TRUTHS and A and E are SNARES.

The beauty of the game is that students do not fear creating a wrong answer. Actually, they like to create answers that are SNARES. Students like to trick their classmates.

Here are some other examples that we have done.

This is what I have been doing when I have my students voting.

This game is a work in progress. Do you have any suggestions for me? What has worked for you? Please let me know.