1A.4:          Well-Structured Lessons

My classes usually began with a low-stakes starter problem that got students thinking and writing about the mathematical concepts we would be exploring that day. I tried to put my students’ names in starters and draw on real-world connections as often as possible to engage students from the very beginning of class. In my numeracy class, starters frequently ended up taking almost half the class because they provoked such rich discussion and debate amongst the students.

In contrast, my integrated math class needed to have more of a sense of urgency when they were doing their work. After realizing this a few months into the year, I started to provide time limits throughout each lesson. Independent thinking time for starters generally lasted one to three minutes, which I found was just enough time for students to produce meaningful thoughts without slipping into side conversations. I also chunked longer assignments so that students had shorter time limits with clear expectations on what should be finished at the end of each time limit. For example, with an activity that was going to take students twenty minutes to complete, I would tell that they had five minutes to complete the first two questions, and then continue to give them new time limits and expectations as the period continued. As long as students were working hard, I was happy to extend the time limit.

Another important component of my lesson structure was incorporating a mix of individual and group work. When I posed a problem to students or asked them a question, I often gave them time to think on their own first, then share with a partner or a small group of students, and then share with the class. I found that this sort of structure not only made students feel more comfortable sharing their thinking with others, but it also allowed them to encounter the problem multiple times from different view points. I generally put students in heterogeneous groups so that they were exposed to different levels of thinking and could learn from each other.

When creating and choosing activities for my classes, I mostly looked for activities that required students to really think, rather than just manipulate numbers. Discovery-based activities, in which students constructed mathematical knowledge for themselves, were my favorite to implement, for they gave students agency over their education and showed them just how powerful their thinking could be. A great example of this was a project my numeracy class did in which students discovered and taught each other exponent rules. Click here to see a video of this student-centered project. I also looked for activities that had some sort of real-world connection so that my students understood why they were learning a certain topic. As I wrote in greater detail in the “Adjustment to Practice” section below, about halfway through the year I began to realize that I needed to incorporate more activities that allowed skills-based practice in order to help my students achieve mastery. However, regardless of the nature of the activity I selected for my students, each of my selections, as well as each step of my lesson plans, had a clearly established rationale. Since I was asking my students to engage in an activity, I felt that it was only fair that I had clear reasons for why I wanted them to do all the different parts of that activity, not to mention the activity as a whole. Click here to see examples of Timed LAP Agendas in which I have extensive rationale sections.

1B.2:          Adjustment to Practice

Throughout this year, I struggled to find the right balance between exploration-based problem-solving activities and skills-based practice. I began the year completely focused on the former, ready to help my students discover the power of mathematics for themselves in one awesome activity after the next. I soon realized that while those activities are indeed powerful, students needed to then do concrete work with the concepts they just discovered in order to reach (or at least come close to) mastery. This became most apparent in my integrated math class as they worked with linear equations. They did pretty well interpreting real-world scenarios and talking about slope and y-intercept in the context of those problems, but then had a hard time writing equations and identifying slope and y-intercept on their quiz in December, since I hadn’t given them much skills-based practice beforehand. As of March, I was still struggling to find a good balance with them while also keeping the engagement high, but I was much more aware of the need for this balance than I was at the beginning of the year.

Since numeracy class is meant to give students support with computational skills, I needed to remind myself to break up the heavy skills-based units with relevant real-world scenarios to keep them engaged. For our distributive property unit, students were getting obviously burnt out from working through so many different worksheets. I understood their frustration with the material, but also knew that the only way they would get better at it was with practice. To mix it up, we started playing games in which they had to simplify expressions in groups, and I also did a series of starters that involved students, the distributive property, and real-world scenarios. This immediately reenergized my students, for we had balanced out the skills-based activities.

Here is an example of one of the starters I created for them:

Kenzie is so excited to buy A Boogie’s new CD! After listening to it online, she decides to buy 3 copies because she loves it so much. Each CD costs d dollars and an additional $0.50 in tax. A Boogie heard through Facebook that Kenzie was a huge fan, so he decided to refund her the money for two of his CDs, not including the tax. To figure out the total amount of money she spent after the refund, Kenzie came up with the following expression: 3(d + 0.50) – 2d, where d is the cost of 1 CD. Use your knowledge of the distributive property and combining like terms to simplify the expression so that it’s easier for Kenzie to figure out how much money she spent. Extension: If A Boogie’s CD costs $9.99, how much money does Kenzie spend?

For both of my classes, I also realized that I needed to do more formal checks for understanding. On a daily basis, I looked at my students’ starters and progress on activities to informally assess comprehension. However, for most of the year I waited a few weeks to give them a large quiz as a summative assessment. My integrated math class especially underperformed on their quizzes in the first half of the year and did not seem to have a solid understanding of the material we had been working with for so long. Starting in mid-March, I began giving them min-quizzes more frequently to check for understanding and encourage them to put more effort into actually learning the material along the way. Visit the evidence and examples portions of my student growth section to read more about my and my students’ growth in terms of assessment.

In addition to making adjustments to my classes the next day, I also frequently made adjustments on the fly. One significant example of this was in February with my numeracy class. As a starter, I had my students create a word problem as a class that involved systems of equations. Students became so wrapped up in the math and posing “what if” questions about the scenario that we spent the entire class working on the problem, and then the entire next day working on a extension to it that the students requested. When moments of rich learning such as this arose, spontaneously changing my lesson to fit the mathematical interests of my students almost always felt worth it. Click on this link to see an illustration of this powerful learning experience.