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


Problem-Based Tasks in Mathematics

Recently, I had the opportunity to visit New Tech High School in Sioux Falls, South Dakota. This school is part of a network of almost 200 schools throughout the United States as well as a few located internationally. These schools strive to teach students through project and problem-based learning. Why? It provides a connection between problems and projects and the real-world. Students who attend these schools do not have to question why they are learning the material because it is applied directly to the real world while they are learning it. So, what is the proof to show that this method of teaching works? Their graduation rate is 92%, which is 9% higher than the national average and students from the New Tech network score higher in academic measures and employment skills. More statistics can be found on their website. However, while I was there I also found out that standardized test scores, while higher in all other subjects, were consistently lower in the area of mathematics when compared to the national average. This made me wonder why there is still such a large push for real world application while teaching math. To help me understand, I found an article, Problem-Based Tasks in Math Deep Dive, about problem-based learning written by Two Rivers Public Charter Schools.

This article talks about the importance of problem-based tasks and how to incorporate them into a traditional classroom. Prior to reading this article, I thought that as a teacher I would have to try and come up with significant problems for every class I taught–which would be very difficult and time consuming. This is not the case. Instead, this article describes using problem-based tasks to introduce new concepts before teaching the students a particular method to solve similar problems. These tasks would only be used about once a week depending on the schedule of the course. On the days that are not used for this task, “lessons focus on building students’ fluency and efficiency in mathematics.” This was great news for me to hear. The article then describes how to structure and efficient problem-based task.

First, before solving the problem, talk about the problem with your students to be sure they all understand the problem. The acronym K.W.I. is used here.

  • What do you Know about the problem?
  • What do you need to find out?
  • What Ideas do you have for solving the problem?

Having students answer these questions provides a strong base for understanding the problem and where to start when solving.

Next, let the students “grapple” with the problem. Let them struggle while you circulate the room, listen to students, and ask probing questions that keep them thinking without revealing too much information. This portion can be done individually, in pairs, or in small groups depending on the level of your students. While in this stage, you should select a few students to share ideas that may spark thought with the other students–remember though, you still do not want too much shared.

And finally, after the problem has been solved, discuss the students solutions and routes they took to solve the problem. Talk about ways that did and did not work. This allows students to view other possibilities and connections they may not have made.

So, as it may sound very difficult and time consuming, problem-based lessons do not have to happen every class. And these guidelines will help myself and you to create more problem-based tasks for your students to struggle with in the future.

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.

Internship Week

This past week I have been on an internship as per requirement of the School of Education. To be completely honest, if I could stay in the classroom the rest of the semester I would. It was so much fun, and is making me so excited for next year on residency! (I just got my placement this week to so it is becoming very real). I had such an amazing experience and had really good interactions with my mentor teacher, his aids and co-teachers, and, most importantly, all of the students. I was in a seventh grade, math classroom at the East Sioux City Middle School.

Part of the internship required teaching/co-teaching a lesson for the class. I taught a review lesson for their third quarter test. My whole lesson was playing two review games which I will describe below. I taught it to four different classes and they all went well.

The first game was a matching game. I had seven different colors of note cards and each color stood for a different sort of question. Each note card either had a question on it or an answer. The students had to solve the question card they had or help a friend with their question. Once answers were found they had to find who had their answer card. Then once an answer and a question card were put together they checked it with me and I told them whether they were right or wrong. The different problems presented on the cards were: fraction multiplication with regular fractions, mixed numbers, and mixing those two, percent of problems, cross multiplication problems, reduction and equality of fractions, similar figure problems (find the missing side), and converting between fractions, decimals, and percentages. It allowed for  a lot of student interaction, participation, critical thinking, collaboration, and problem solving.

The next review game was a Kahoot game: I created it explicitly for my internship class. This game went over really well because the students knew what was Kahoot was already. They really enjoyed being able to compete.

I learned so many lessons from my week of internship. Behavior management goes over a lot better if you are consistent with the rules that you are following and be fair with students. Always stay positive and do not make kids feel like a burden when they come to answer questions. Students generally would come and ask me questions a lot because I was always looking around and walking around the classroom. In one class I taught a girl was overwhelmed with all of the movement and noise, so I made sure to be sensitive of that and allowed her to move to an adjacent classroom to ours and she worked quietly in there.

Overall, I could not have asked for a better internship week and I am sad to be leaving it. It has made everything coming up next year so much more real and I could not be more excited to be in the classroom long term.

Let’s Talk About Math

When planning a math lesson, teachers often try to think about how to engage their students by using examples from the real world aligned with the students interests, group tasks that force students to work together to complete a specific problem, or the basic question and answer styled lesson. Rarely do they think about having a class discussion…but why? Why is it that math classes shy away from having discussions to teach content? One reason might be that it is difficult to gear questions that promote student interaction and discussion.

Students who talk about math are able to address gaps in their understanding of mathematical concepts and allow them time to express their ideas more precisely. After reading the article Talking Math: How to Engage Students in Mathematical Discourse, I have found a few tips that will help promote this type of classroom.

First, be sure that your students are aware that you expect them all to participate in the discussion. You cannot run an effective discussion if only two students do all the talking for the class. Make sure everyone feels comfortable–starting this with small groups may alleviate anxiety for some students. Second, discuss errors and solutions in detail. Be sure that each student understands the reasoning behind each solution. Third, go over definitions and introduce new mathematical vocabulary during lectures and general conversations with your students. Setting students up with the right words will make them more confident when they are talking about math in front of the group. And most importantly, let the students talk! If you hear a student say something that is wrong, let them talk it out for a bit. See if your students can help each other before you chime into the discussion. The following image shows what you might see in a classroom that has meaningful mathematical discourse.


Mathematics in the Classroom or the Real World

Throughout grade school we always heard our math teachers say things like “math is so important and will be used in almost everyones daily life.” But they rarely showed us to what extent this was true. I could see using estimation when shopping or ratios when using models but for other math lessons, it was often very difficult to determine where exactly one might use this in real life. Studies have shown that students are more engaged and motivated to learn math if they can see a reason and use for the material. After talking about 3-ACT math tasks in class, I was trying to think of more time-effective activities that students could do to see math in daily life. I came across an article called The Educator’s Guide to Applying RrReal-World Math: 15 Resources with over 100 Lesson Plans, written by HomeAdvisor.

This article talks about using connections between math and the real world for kindergarten through high school. They discuss having students drawing scaled models of their classroom using ratios and proportions and use probability for gambling situations. As a teacher, coming up with real world application problems for every lesson can be a challenge. This website gives applications for many lessons that are very helpful in engaging students in the material. It also discusses how lecture is important to teach certain concepts.

Geometry within a Bridge

Using Learner-Generated Examples to Increase Understanding

Learner-Generated Example (LGE): Examples of concepts that are created by students

In the majority of mathematics classrooms, teachers use what is called lower-order thinking to teach their students. Some verbs that are in Bloom’s Taxonomy, which shows what’s higher- or lower-order thinking, that are a part of lower-order thinking are words like calculate, solve, apply, complete, produce, and manipulate. These are all words that teachers use after teaching a lesson and giving students a large amount of problems and tell them to do finish them. This approach is fine. Practice does help. However, if we want our students to have a deeper understanding through the use of higher-order thinking, we should use activities that use verbs like create, compose, invent, and formulate. One way to promote this higher-order thinking is by having students create their own examples and interpretations of concepts, called Learner-Generated Examples (LGE).

I learned about Learner-Generated Examples from an article in NCTM’s Mathematics Teacher titled “Learning about Functions Through Learner-Generated Examples” by Martha O. Dinkelman Laurie O. Cavey. The article describes how they used a multiple-step process of formal formative assessment using LGE. They started by teaching about functions. After this step, they had students list everything they knew about functions. They worked up to having students give an example of a function in four different formats. They later came back to this method after seeing what they needed to teach and teaching it to the students. This is a great use of LGE, because students may not be able to put into words what they know about functions, but having them give examples of functions shows what their preference is. The teachers learned that while students could have written out the equation, drawn graphs, or written what the function did in words, students usually just wrote a series of numbers for x-values and y-values. This showed that most students thought of functions and x- and y-values, which meant that the teachers had to work on helping them understand all the forms of functions.

In our 7-12 Math Methods class and our Curriculum and Instruction for Middle/Secondary School class we’ve learned about the value of having multiple correct answers. If there’s different correct answers, that means that there’s multiple ways to come up with those answers, which provides an opportunity for students to struggle and collaborate. There is certainly multiple correct answers when students have to come up with their own examples. For an example of an example, if students have to give examples of how a difference in slope in a linear equation changes the appearance of the line on a graph, there’s an infinite number of potential answers. They also have to identify what number in an equation symbolizes the slope and make the connection between the equation and the graph.

Another useful part of LGE is that it can be used for many different concepts. Students can create their own graphs. They can create their own equations. They can create tables or dimensions of objects or shapes. They can also use technology to create these examples. Students could draw a random line and try to figure out the equation of it. This is a great way to use Desmos. Students can experiment with entering equations and seeing what shapes they can create.

The way that I would use this in my classroom is as a formative assessment to check and see what students have learned and what I need to work on. I like the idea that the article had about asking students to provide multiple examples for a single function. This can show what they’ve learned and what they need to work on. It can help me as a teacher to figure out what my weaknesses are in a lesson. Most importantly, I think it deepens students understanding of a concept by having them create an example that fits the main idea.

Link to NCTM article:

Learning About Functions Through Learner-Generated Examples