Why You Should Start the Year with a Unit 0?
What is a Unit 0?
A Unit 0 is what you teach before you start teaching. It’s what sets the stage for the culture of the classroom and the expectations for learning for the rest of the year. Read on to discover its benefits and how it can look in a mathematics classroom.
Why a Unit 0?
My Experience as a Math Teacher
As a math teacher content was king! I had a short time to get through a ton of material before state testing, and we couldn’t waste a day. So Day 1 was syllabus day, and then on Day 2 we dove deep into content. There was no time to waste! To be fair, I would start the year with hands on engaging, cooperative activities like my Power and Exponent Exploration with Cube Roots activity which can be found on my Math Lessons page, but I didn’t have time for anything that I would have considered superfluous or a distraction from content.
My Experience as a State Lead for Ohio’s Mathematical Modeling and Reasoning Course
My Role
When working for the Ohio Department of Education (now named the Ohio Department of Education and Workforce), I was an education program specialist. One of my projects was to work with Steve Miller from Summit County Educational Service Center on a new high school Quantitative Reasoning course that would later count as an Algebra 2 equivalency course. We titled the course Mathematical Modeling and Reasoning. This course was meant to prepare students for the Quantitative Reasoning Pathway (and other pathways) at the postsecondary level. It was completely different than a traditional math class!
Note: Other states have been going in a similar direction as Ohio by developing high school pathways. For more information on the Ohio’s High School Math Pathways view the Ohio Department of Education website.
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The Pilot
A group of expert teachers and higher education faculty came together to write an inquiry-based course focused on problem solving and questioning. All the problems were based in real-world contexts, and there was an emphasis on communication. While creating the course materials there was constant communication between the writers and other stakeholders including postsecondary math professors. This was accomplished through an advisory council, working with the Ohio Department of Higer Education, and updates to the Ohio Math Initiative.
Additionally, the pilot teachers were required to do an extensive ongoing professional development turning them from inexperienced novices in the course to teacher leaders for upcoming cohorts.
Observations
During the first year of the pilot, Steve and I did classroom visits to see how the course was going. We interviewed teachers and students for feedback. Both the teachers and the students loved the course! Students were learning a lot; they were engaged and having fun! Because of the emphasis on critical thinking and questioning, teachers were claiming it was more rigorous than their Pre-calculus class.
Since I wanted feedback on lessons, I asked the students to describe their least favorite lesson in the course. I kept hearing a similar theme across the state. Students kept telling me about lessons that weren’t officially part of the course, which was odd. We finally deduced that these were problem solving lessons that teachers gave students the first few weeks to set the stage for learning, which we encouraged them to do.
Solution and Response
After our observations, Steve and I came back to convene. Since setting the stage for this type of learning is crucial, we realized that we needed a specific structure for the beginning of the year. It wasn’t content that the students were missing, but they need to learn how to learn in this new way. And the teachers needed more explicit support on how teach in a different way.
We needed engaging low-entry, high ceiling activities that would establish a classroom structure that teachers could use prepackaged straight out of the box. It needed to focus on questioning, collaboration, our routines, and the mathematical practices. Since there was nothing we knew out there that existed, we created one! Hence the birth of Unit 0 (or as we called it Theme 0).
When we rolled it out the next year the teachers loved it! They said that having a Unit 0 (or a Theme 0 as the course calls it) was a game changer in terms of implementing the curriculum.
Description of Ohio’s Mathematical Modeling and Reasoning’s Unit 0
What we did in our Unit 0 would be applicable to any math or science classroom that wanted to emphasize problem solving and inquiry-based learning grounded in real-world contexts. As you can see by the course’s scope and sequence, the teachers spent 2-3 weeks setting the stage for the rest of the course.
Standards of Mathematical Practice
One of the main things we did was that we explicitly taught the Standards of Mathematical Practice. We would choose an engaging mathematical task that would highlight the specific practice. After completing the task, we would have students identify which of the mathematical practices we used, and then we would have them identify where they were on the Math Practice Rubric. This drew explicit attention to the mathematical practice, and it showed them how they could grow in the future.
Here are some websites that may give you some ideas on how to choose and implement tasks surrounding the math practices:
Evaluation of the Standards of Mathematical Practice
As you know, students only take seriously what they are evaluated on. So, we asked ourselves: how can we evaluate the standards of mathematical practice? The answer was that we created a Standards of Mathematical Practices Rubric. It could be used for student self-evaluation, or teachers could use it for a formal evaluation. We would suggest only focusing on one or two math practices per lesson.
Although this rubric targets high school, elementary and middle school teachers could adapt the concept and create something similar for their classrooms.
Mathematical Mindset: Fixed vs Growth
Carol Dweck
Because many students have a history of struggling in math, we wanted to change their mathematical mindset. Carol Dweck wrote a book called Mindset: The New Psychology of Success. In her book she talks about the difference between a fixed mindset and a growth mindset. She claims that focusing too much on intelligence can make a person have a fixed mindset. A fixed mindset is where one believes their abilities are fixed or carved in stone and cannot be changed. Whereas a person with a growth mindset believes that you can cultivate your intelligence through your efforts. The idea is that a person’s true potential is unknown and cannot be predicted, but people who put forth more hard work and effort will see greater success than those who perceive that their intelligence as fixed. People with growth mindsets will also take more risks and try more things which will cause their intelligence to actually grow.
Jo Boaler
Jo Boaler built on Carol Dweck’s work and wrote a book called Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages, and Innovative Teaching. She emphasizes the importance of giving students rich tasks where they are forced to struggle. Mistakes are a powerful learning tool that can be leveraged to increase learning; therefore, it’s important to create a classroom where it’s safe to make mistakes.
Mathematical Modeling and Reasoning and Mindset
Therefore in our Unit 0, we focused on choosing tasks that forced students to struggle productively, preserve, and were prone to making mistakes. Then we leveraged these tasks to have discussions about why these qualities are important in a math classroom.
If you are looking for activities to use in the first few weeks of school to encourage growth mindset, Jo Boaler’s book has a lot of great activities in the appendix that could be used in the classroom.
Norms and Routines
Another thing we did in Unit 0 is that we set classroom norms and practiced routines that we were going to use in the classroom throughout the year. Since collaboration and communication are vital for a math class that focuses on problem solving, on Day 1 we gave students a fun non-mathematical task that forced them to collaborate and communicate.
Then as the week progressed we added in more mathematical tasks while introducing other routines that we would use throughout the year. Some of the routines we used were Notice and Wonder, 3-Act Tasks, Modeling Routines, Number Talks, and Fermi Problems among others.
Notice and Wonder
Notice and Wonder is a simple routine that sparks math engagement; creates a safe space for discussion; helps students look into details; and helps students form their own questions. Just give students an engaging problem or a mathematical picture and ask students what they notice and wonder. You’ll be amazed at the results this simple routine and language provokes in conversations. This routine and the noticing/wondering language that accompanies it is useful for a variety of situations outside of mathematics as well.
Here are some resources to help you implement the Notice & Wonder routine:
- MPIR – Notice and Wonder | OER Commons
- Notice_and_Wonder_Process_and_Resources_4.25.19_AiKcYJD.pdf (oercommons.s3.amazonaws.com)
- Notice/Wonder – Formative Strategy (smartertoolsforteachers.org)
Number Talks
Number Talks is a short routine that helps students develop number sense. The teacher presents a problem, and the students solve it mentally. Then students share different strategies for solving the problem making sure to explain their thinking. Oftentimes teachers start with dot talks, and then progress to number talks. There are also resources for fraction talks and data talks.
Here is a handout on how to implement Number Talks in the classroom: Number-Talk-Process-and-Resources.pdf (asdn.org)
Here are good websites to find examples of Number Talks:
- WIM-Dot-Card-and-Number-Talks-Grades-K-12.pdf (youcubed.org)
- Dot Card and Number Talks (activity) – YouCubed
- Dot Number Talks (mathsplay.org)
- Number Talks – Math For Love
- Number Talks for Middle Schoolers — Fawn Nguyen
- Fraction Talks – Math For Love
- Data Talks Archives – YouCubed
Since understanding fractions are a predictor of success in mathematics, we developed a Number Talk Scope and Sequence for MMR aimed at high school students around learning fractions.
If you want to take a deeper dive into Number Talks, here are some books with plenty of examples that will help:
Three-Act Tasks
A Three-Act Task is an interesting mathematical situation that is shown in three parts or acts. Students focus on questioning, attending to details, and oftentimes estimation. Act 1 shows the context for inquiry. Act 2 gives more information, and Act 3 is the big reveal. The beauty of the Three-Act Task is that it is engaging, and it forces students to pause and think.
Here are some good websites for Three-Act Tasks:
- Dan Meyer’s Three Act Tasks | Math (newvisions.org)
- The 3 Act Math Beginner’s Guide | Spark Curiosity to Fuel Sense Making (tapintoteenminds.com)
- 3-Act Tasks | Questioning My Metacognition (gfletchy.com)
Modeling
As you can decipher by the name of our course, modeling mathematics was the center of our Mathematical Modeling and Reasoning course.
The Common Core has as diagram for the modeling process it promotes, but it is can be hard for students to remember.
The GAIMME (Guidelines for Assessment and Instruction in Mathematics Modeling Education) Report is one of the more authoritative resources for modeling in the classroom, and they also have a modeling process. It’s a must read for incorporating modeling in the classroom!
However we thought Robert Kaplinsky’s Spies and Analyst Model was friendlier for students. So we took his idea, incorporated some of the elements from the GAIMME report and made a modeling routine for the students. I don’t think the worksheet we created on this is public facing yet, but you can either make your own or contact Steve Miller and Summit County ESC, and he might share the handout with you.
Communication, Conjectures, and Proof
To be mathematically proficient citizens students need to be able to justify their reasoning using quantitative information, and they need to be able to communicate it clearly, so we also added these elements into our Unit 0 as well. We created a lesson that incorporated some of the common misconceptions/fallacies that students make when trying to prove things and confronted them head on.
So Why a Unit 0?
A Unit 0 teaches students to think mathematically. By incorporating a Unit 0 into your curriculum, your students are learning mathematical habits of mind that will last them not only the rest of the course but for the rest of their lives. You are setting the stage to train students to become lifelong learners and questioners. Students will be able to articulate the math practices, apply them in new situations, and become more fluent in using them. I know it’s hard as a math teacher to give up any time teaching specific content standards as that state test is looming in the distance, but by including a Unit 0 you will reap the rewards as students learn how to learn. This will allow them to have greater retention of what you teach them (or they discover) as the year progresses. If you’re worried about not having enough time for your content, try double-dipping; find Unit 0 type tasks that hit some of your grade-level content standards as well as introduce routines and math practices.
Just a Warning
This may change your life! It’s important to know that if you introduce students to learn this way, you as a teacher need to keep the trajectory going and not fall back into old habits of traditional teaching. Starting with a Unit 0 may forever change both you and your students as the math classroom and culture becomes exciting instead of stale.
“If you always do what you’ve always done, you always get what you’ve always gotten.”
~Jessie Potter
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