Setting the Stage for Productive Struggle

A growing interest in teaching with productive struggle is a hopeful sign in the math classroom. Productive struggle can lead to deeper understandings, connections, and motivation in students. So, why are some teachers still hesitant to use it?

Recently, I was able to present a lesson over the Law of Sines and Cosines in a local high school. Normally, these students learn in the traditional way of lecture. However, we challenged ourselves to create a lesson that caused students to struggle a little before they came to the solution. When we presented it, many of the students seemed disinterested and unengaged. I was worried about this when creating the lesson, but I did not think it would occur to the extent that it did. So, I have looked into articles that provide ways for a teacher to slowly introduce and transition students into lessons that incorporate productive struggle.

The article, Beyond Growth Mindset: Creating Classroom Opportunities for Meaningful Struggle, gives tips on what to do and what to avoid when teaching with struggle. Years of research has proven that student learning is enhanced when they have to be persistent to reach success–this concept goes all the way back to the educational reformer, John Dewey who “described learning as beginning with a dilemma.” One major key to a successful productive struggle lesson is to have the goal of the lesson be focus on getting students to have a deeper understanding of the material instead of just focusing on creating struggle. One common way to do this is through real world applications. For example, in the lesson we created for the geometry class, there was an activity that involved drones and using them to deliver pizza. Although this is not necessarily happening today, it is something that could take place in the near future. When this activity was introduced towards the end, there seemed to be a switch that flipped. Almost all of the students were working together to figure out the problem. And, they were able to recall previous math knowledge to help them solve the Law of Sines problem. This is another helpful tip when causing productive struggle–having students use math that they are familiar with can help produce productive struggle instead of frustration. Having the knowledge needed for part of the problem gives students a boost of confidence. However, students also need the time to realize these connections. As a teacher, you should be sure to give your students enough time to think about the activity before giving them any hints or guidance. This time is crucial for allowing students to make connections and deepen their understanding of the math at hand.

The article provides the following list of key elements for providing productive struggle:

  • Determine timing and placement for productive struggle within the unit or curriculum—lessons that are “preparing students to hear something really important.”
  • Align struggle activities with clear, specific learning goals.
  • Design struggle tasks based on assessment of students’ prior knowledge and skills.
  • Foster a safe environment that encourages student inquiry and exploration of important ideas.
  • Use probing questions to solicit student thinking and provide strategic assistance to nudge students through their zone of proximal development–the zone of students’ thinking just beyond the level they can do completely on their own.
  • Follow-up each struggle episode with carefully structured lessons that build on students’ ideas, address misconceptions, and help students forge new understandings.
  • Assist students to reflect and articulate what they learned as a result of productive persistence.

In the future, these tips will definitely help to transition into a classroom with productive struggle much better than the geometry class I just experienced. For instance, I think that the students would have gotten much more out of our discovery activity for the Law of Sines and Cosines if we gave them more thinking time and created an environment that allowed students to talk about their ideas without fear of being wrong.

Image result for productive struggle in the classroom


Yummy Math That Doesn’t Involve Eating Pie on Pi Day

If you were hoping for excuses to eat food while teaching math, this probably isn’t you blog post to read. Instead, I am writing about Yummy Math, which is a website that provides interesting lessons for students of all ages. While not the most visually appealing website, it is very useful for a math teacher looking for resources. . The lessons they have are about things that are useful or realistic (unlike the problems about buying 60 pumpkins or eating 20 candy bars). They are also aligned to standards. They have tags on every problem with the standards they are aligned to. There is also the option to search for lessons based on the standards you want to address. The lessons go from 2nd grade mathematics to high school. A one- year membership only costs twenty-two dollars and gets you access to all the materials that they have on the site.

As someone who will be student teaching next year, I’m always on the hunt for valuable resources to help me teach. I think that one of the best ways to do this is to find websites like Yummy Math. It can be a challenge if one wants lessons that are geared towards standards, challenging, and interesting to students, but we have to put in the work as teachers. It’d be great if we could come up with really fun yet challenging tasks for our students on a daily basis, but the truth is that there just isn’t always time in the day, especially if you have to teach five or six different classes each day. This is why great resources are important, and I’m happy to say I’ve found another one.

Motivating Students in Math

Recently, we have been discussing ways to make students more motivated in math because for many interest is lost between elementary school and high school. This interest is generally not regained until college or beyond. Janelle Cox wrote an article, How to Motivate Students to Love Math, that teaches teachers how to get students to be more interested.

One of the first things she discusses is building on skills that students have already mastered. Whether this be something as simple as addition or as complex as the Pythagorean Theorem, letting the student use knowledge they feel comfortable with will make them less frustrated and more likely to remember the new connections they make. Then, work towards an achievable and personalized goal. When students meet one goal, they are more likely to strive for more and also more difficult ones. So, be sure the first goal they set is somewhat short-term and achievable–you may have to help them with this. Incorporate things that pertain to their life into the lesson such as real world examples and applications as well as the use of technology. Some of this may mean you have to do a little research and learn the latest technology and websites for math, but it will benefit your students tremendously.

And finally, make math interesting! It does not always have to be lecture and learn. Use music, games, and be enthusiastic when teaching and discussing math. Students pick up on your tone, so if you are bored teaching and do not seem to have a passion for the content, they will not be interested either. One more way to intrigue students would be with a magical math problem. This means finding math in the real world that might give an unexpected or interesting result. As we have talked about in class and the article talks about, the birthday problem. This problem talks about the probability of two students in the class having the same birthday. Proving this in your own classroom may be the one thing that sparks an interest in math for your students.

While it may be difficult to be creative with teaching every lesson, incorporating as many motivational lessons as you can will both show your students you care and keep them engaged. These ideas and tips will help create a more effective teacher and get students more interested in math and it’s applications. InspirationalMathQuote3.1-Instagram.png

Silent Discussions

Some major goals of mathematics are to have a deep understanding of the content and to be able to make connections to other concepts. I decided to look around the NCTM website for articles about such things, and I found one called “Visible Thinking in High School Mathematics.” This article is about two main methods: Chalk Talk and Claim-Support-Question. I’m going to focus on Chalk Today, because it really caught my interest.

The main idea is to have a variety of posters around the room with questions on them, generally sounding something like “What do you know about (concept).” Forever however many posters there are, say five, that many different colors of markers are distributed among the students. Students with all the same color markers are sent to a poster, and are told to write what they know about the concept. This is a totally silent activity, which is why the author called it “Silent Discussion.”Students then rotate around the room and either respond to what other students wrote or write their own new idea.

Chalk Talk gives students the opportunity to look at other students’ ideas and get their questions answered at least partially by other students. For a question such as “What is a quadrilateral,” a student may have thought of a square, but with Chalk Talk, they can get the opportunity to see a non-square rectangle, rhombus, parallelogram, or any other quadrilateral, possibly with a picture and description. It gets them thinking outside the box. If they aren’t sure about something, they can ask, and the next group at the poster won’t even know who wrote it, and they can get an answer for their question. That’s ideal.

The posters really end up looking like a mess, but the teachers can somewhat gather what the class knows and doesn’t know, as well as where the class should go next. Even if questions get answered, it still shows that students might not quite feel comfortable with a concept. On the other hand, a question asking about a possible future direction from their new knowledge can make for a great transition into the next topic. Also, students enjoy getting out of their seats, and this is a productive way to do that. It is a great idea overall, in my opinion.

Visible Thinking in High School Mathematics

There is No Perfect Lesson

So, we know the Law of Cosines is taught in many high schools, but how many of you could truly explain the Law of Cosines right off the bat to someone who had never been introduced to the content? It would be difficult. Throughout the passed few weeks, we have been conducting lesson studies on how to teach this material as well as a few other topics. Many of the searches lead to a dead end of endless worksheets that just force the student to repeat the formula over and over again. Even when learning it for myself, my teacher had me practice the standard problems 1-50 after giving an hour lecture that showed us how to use the formula or memorize the formula with a song.

As teachers, shouldn’t we be working towards an environment that produces mathematicians who work to solve problems instead of just memorizing meaningless formulas? This problem is often run into with various lessons in mathematics, which is why lesson study is so important. When we began studying how to teach, the first thoughts that popped into my head were things like “Oh this should be easy, google has everything” or “someone had to come up with an effective and engaging lesson to teach this already.” But this is not true. Many ‘good’ lessons fail to exist even with all of the current research going on.

While researching, I found myself not only looking at how to teach the law of cosines but also how lesson study improves student learning. An article from Classroom Chronicles did an amazing job of explaining the process and just how important lesson study is to the future of education.

Lesson study is the first step in turning a traditional style classroom with lecture into a classroom where students explore and discuss cognitively demanding tasks. The process of lesson study begins with multiple professionals working together to create a strong lesson for students. This will require a lot of research and time to determine what the best way to teach students would be. Next, teachers teach the lesson and then observe how the lesson went. This can be done in a variety of ways such as having another teacher sit in and give feedback and setting up cameras to record the lesson for your viewing later. After reviewing the lesson, ask and answer questions with other math teachers about your lesson and it’s pros and cons. Then, revise. Change the parts of your lesson so it runs more smoothly, do not just put it off until next year when you have forgotten the small flaws. This process is then repeated year after year to modernize and update the lesson for better student engagement and understanding.

The largest benefit I can see from lesson studies is the overall improvement for the future of education. You are creating and working on lessons to better serve students and their needs and providing these lessons to other teachers who can impact even more students. After I begin my teaching, I plan to begin a blog with lessons I have worked on with coworkers and make it accessible to other teachers–allowing them to post their lessons as well. This will create a strong base for teachers who are new to the field as well as teachers who have been struggling to reach students with the traditional lecture, as there is always room for improvement.

Not Only Should We Use Real-Life Examples, but We Should Choose Real-Life Examples That Target Interests

In order to motivate students and deepen their understanding of math concepts, we as teachers/future teachers should use real-life examples. If the only problems we ever use in our classes are basic problems consisting of numbers and variables, students will get bored very quickly. Many textbooks include problems that could occur in real life, but these problems are rarely ones that students would do on any given day. They may include buying 40 shirts or 60 pumpkins. I distinctly remember doing a problem where we figured out the length of a guy wire when given the length of a telephone pole and how far away from the pole the guy wire touches the ground. I can’t remember the last time I figured that out in real life. One of my professors preaches that the real-life problems we do actually need to be things we would do in real life. I wholeheartedly agree, but I’d like to take it a step further and suggest that we do problems that students would want to do in real life, even if they haven’t thought of it.

The first step is keeping up with students. Right now lots of people, especially boys, are obsessed with Fortnite. Many more enjoy sports. People obviously love social media. Teachers need to take the time to get to know their students’ interests so they can create lesson plans around these interests. Next is identifying numbers within these interests. This could be numbers of calories, numbers of followers, completion percentages in football, etc. Then problems that challenge the students need to be made. It needs to be at the correct level of difficulty while also deepening understanding and encouraging discovery. Finally, the teacher needs to find a way to smoothly fit these problems into the curriculum. They should not be random problems using concepts from weeks or months earlier.

A great website that has such problems is Mathalicious. It has 135 challenging problems for all types of math concepts that help students find out interesting facts. One problem that caught my eye was about basketball. It involves finding out whether fouling an opponent on a game winning shot is a good idea or not. It has students find the probability of each team winning and losing in either scenario in a real life situation. It really interested me, because I like sports. There are also other problems about social media, food, tv shows, games, fact about the world, and much more. It’s a great resource for finding problems that interest students.

My Internship Week

This past week (12th through the 16th), I completed my secondary education internship. In case you don’t know I have a double major in secondary mathematics education (7-12) and special education (K-12). This internship was for the secondary math education major. I was at North Sioux City Middle School from 7:25-3:05 every day for a week. I learned many lessons during my time in the classroom, but a few I’d like to focus on are: Having a backup plan, making sure everyone understands, and that there are many resources out there.

The first lesson I learned is to always have a backup plan. This is especially true when it comes to technology, since it seems to fail so often. Three out of the four days I was in the classroom, something came up that forced my mentor teacher to change her plans. One day (Wednesday) it was the walkout that took place. Another time it was just that many students had a school activity going on that she wasn’t told about. The third time, the computer program she wanted to use wasn’t working properly. Every time, she seamlessly switched to another lesson. I’m not even sure the students noticed that she had changed her plan. That’s because she always had a backup plan. On Thursday of my internship week, I taught a lesson that required the students to use Desmos on laptops. Logging in to the laptops took at least five minutes, and since I had not planned for this set-back, all I could do was stand up front and ask whether they were logged in. After my lesson, when my mentor teacher and I were talking, we discussed how important space-fillers and backup plans are when it comes to technology. That’s another takeaway, along with just having a backup plan in general.

My second major takeaway is to make sure everyone understands. One thing that I witnessed, and this doesn’t really hurt anyone except the teacher, was that my mentor teacher would plan for the fastest students. I did the same thing when I did my lesson plan. When I was trying to figure out how long it was going to take, I was thinking about how long it took the few fastest students to do a similar task. In reality, learning takes time. Checking whether everyone understands takes more time than I would have thought. A teacher needs to go around and make sure that everyone knows what’s going on, so everyone benefits from the lesson.

My third takeaway is that teachers use a lot of different resources. There’s definitely many teachers that don’t, but there are so many teachers sharing what works for them. Why wouldn’t a teacher want to find lessons that work. On a daily basis, my mentor teacher was trying to find more lessons that work, and she ended up finding more interactive, life-like lessons for things she was going to teach in a less fun way. One collaborative sort of website I found was In this website, one can search for a standard in a grade or class and find lesson plans for that standard that other teachers have shared. There seems to be a limited number of people that have shared their lessons, but it seems like a great idea. Searching the internet for lessons that can get students more interested in mathematics in worth the time.

Other notes:

It was really interesting to finally see things we’ve learned in class implemented by real teachers. For example, if someone had headphones in or their phone out, my mentor teacher would go up and whisper to them to put it away, rather than yell or pull it out of their hand or anything like that, which is something we learned in class.

It was a new experience to see the break room at the school I was at. If there’s a reason to not become a teacher, I heard it there. They complained about pay, parents, behavior, etc. It makes sense, though. Teaching is a tough profession and at some point, we all need to vent. On the other hand, it was awesome that my mentor teacher and a few other teachers told me they wouldn’t trade their job for any other one. That was a really cool moment.

Middle school is an odd age to me. In some ways, they don’t even seem close to adults. They don’t take care of their hygiene, they behave like middle-schoolers, and they love games and throwing things. On the other hand, some have the same interests as me. There’s also always a group that thinks they’re too cool for me. I’m not saying I don’t like it, it’s just an interesting age.

Those are my takeaways from my internship. I definitely learned a lot, so it was a very valuable experience. It’s great to get out of the college classroom and get some time doing what I want to do the rest of my life.