Chapter 1: The Tasks We Use In Class

How I Understand This Chapter:

The types of tasks we assign to our students are important in building their ability to think critically and problem solve. Liljedahl advocates for the use of tasks that “require students to invoke their knowledge in ways that have not been routinized” (pg 20). He labels these as “Highly Engaging Thinking Tasks”, which he further breaks down into three types: Rich Thinking Tasks, Card Tricks, and Numeracy Tasks. The problem, though, is that many students will shut down or not even try if they have not been primed to think this way. He advocates for using “Non-Curricular Tasks”, such as the Tax Collector task, at the beginning of the school year to engage students in the type of work and thinking they will be doing in the classroom. The goal is to spend 3-5 class days at the beginning of the year having students work on non-traditional thinking tasks that may not even be aligned to a given content standard for the grade you teach in order to train your students to think deeply, look for patterns, and to know that they can persevere through challenging math that has not already become routine for them. Once this mindset and class structure has been formalized, you move toward “scripted curricular tasks” that get students thinking about the math you need to teach them during the school year.

Impactful Quotes (To Me):

“…problem solving is what we do when we don’t know what to do. That is, problem solving is not the precise application of a known procedure. It is not the implementation of a taught algorithm. And it is not the smooth execution of a formula. Problem solving is a messy, non-linear, and idiosyncratic process. Students will get stuck. They will think. And they will get unstuck. And when they do, they will learn – they will learn about mathematics, they will learn about themselves, and they will learn how to think.” (pgs 19-20)

This message stopped me for a while as I thought about all of the times I gave my students “problem solving tasks” that were really just rote algebra problems re-written as word problems. It’s not really problem solving if you can apply a simple previously known algorithm or procedure to something once you decode the words. Turning the equation 2x+7 = 12 into a story problem isn’t problem solving. If this is the only exposure students have to “problem solving”, they aren’t really thinking, and the math isn’t going to be very engaging or interesting. How can I expect students to struggle and persevere through novel, rich tasks when all they have ever seen is rote algebra in a new delivery system?

“Mimicking is an addiction that is easily acquired at lower grades and difficult to give up at higher grades.” (pg 30).

I feel this quote so hard. Even though I talked about it in the introduction section, the words “studenting” and “mimicking” exemplify so many math students in middle school. If I can use these strategies and teaching practices to break even 25% of my students from their “studenting” and “mimicking” addiction, I consider this a complete victory.

“Well selected non-curriculum tasks, with their highly engaging contexts, propel students to want to begin to think. They create situations where every student gets stuck, which makes stuck an expected, safe, and socially acceptable state to be in. In essence, these tasks make it safe to fail and keep trying.” (pg 31).

I can’t begin to express how important it is for students in a classroom to feel fine about failing and not always knowing what to do. For so many reasons students in a math classroom feel they must always be right the first time, and that being wrong isn’t ok. Students need to feel safe to try things, be wrong, try other things, still be wrong, and then try something else. I always joke with my students that if you give them a video game they will probably fail 8 million times and be fine with it, but if they get one math problem wrong it’s a national emergency. One of my constant mottoes in class is “Every wrong answer gets you closer to the right one!”

How Did This Go For Me?

The first week of school can be a real challenge, trying to teach classroom procedures, establish behavior expectations, get to know students, and begin to build a positive, safe classroom culture. With all of that in mind, I decided to go full force into the “Non-Curricular Tasks” for the first six days of school. Each day my goal was to cover four things during the 51-minute period:

I wanted students to know that when they came into the classroom, they needed to be ready to think, talk, and persevere for 51 minutes without a break. Everyone would be expected to contribute, and sharing ideas (both ones that worked and ones that didn’t) was a treasured practice in the room. I wanted to emphasize thinking, and limit studenting and mimicking as much as possible.

Here’s a basic rundown of what I did in each of the first six days of school.

First Day of School (Friday):

  • Attendance
  • Review the Bathroom Pass procedures and where students keep their backpacks.
  • Which One Doesn’t Belong: 
  • Four Fours Thinking Task: This task was done in random groups of three students at the standing whiteboards.

Second Day of School (Monday):

  • Attendance
  • Fill out student planner and enroll students into DeltaMath.
  • Which One Doesn’t Belong: 
  • The Tax Collector Thinking Task. This task was done in random groups of three students at the standing whiteboards.

Third Day of School (Tuesday):

  • Attendance
  • Introduce procedures for borrowing supplies and expectations for how class supplies are treated.
  • Visual Pattern: 
  • Number Palindromes Activity. Since this requires paper and different colored markers, I had students work at the whiteboard tables in pairs. I could not figure out an efficient way to do this lesson on the VNPS.

Fourth Day of School (Wednesday):

  • Attendance
  • Establish Norms for class discussion and sharing ideas
  • Number Talk: 48 + 63
  • The Farmer’s Dilemma Thinking Task: This task was done in random groups of three students at the standing whiteboards.

Fifth Day of School (Thursday):

  • Attendance
  • Review the Classroom Success Plan
  • My Favorite No: Solve the equation 8 – 3x = -1
  • Guess My Rule game: Present students with a T-Chart that has two (x, y) pairs filled out. Students must use mental math only to figure out the rule being used. When a student thinks they know the rule, give them a random x-value and have them predict the corresponding y-value. After 5-6 pairs have been revealed, have a student explain the rule in detail. 

Sixth Day of School (Friday):

  • Attendance
  • Get assigned textbooks from the media center
  • How Many 7’s Thinking Task: If you write out the numbers from 1 to 1000, how many times will you write the number 7?

Without a doubt, this was the best first week of school I have ever experienced. By day four the students were excited to get working at the whiteboards and I saw almost 100% engagement from the students during the thinking tasks. I heard many comments from students at the end of class that their “brain hurt, but in a good way”. A good thinking task has a low floor, but can also offer challenging extensions for teams that want it. Even the most fragile middle school math student can add four fours to make the number 16, giving them a solid start on a task that can get quite complex.

One other benefit that I was not anticipating was that my voice didn’t hurt at the end of each day, since I was doing much less talking using my projected teacher voice. Once the tasks were explained, the students had to figure them out and I was mostly monitoring behavior and maybe giving a hint here and there (more on how to answer student questions in chapter 5).

I was anticipating more push-back from my Enhanced Math 1 students, since none of the work actually covered the standards for the course, and those students are usually the most motivated to learn new material. What I found was that they really dove into the problems, probably because they were a different kind of thinking than they were used to. Accelerated students are often very fluent in using memorized algorithms and procedures. The tasks presented didn’t really rely on those, so they had to think differently. They also hadn’t done those tasks with their tutors already, so they were novel and challenging.

After having such a successful first week of school, I was worried about how class would go once we got into the actual math standards for the courses, and how I would make those lessons as engaging and “brain hurt in a good way”. Finding interesting thinking tasks online about patterns and number theory is pretty easy. Finding or creating rich thinking tasks about solving two-step equations with rational numbers? That’s another story.

Next up: Chapter 2: Forming Groups