One (Kind) Kindergarten

Hello and welcome back to cycle three of my posts on teaching mathematics in a task-based classroom! This week, I will be talking about how the "formative five" allow teachers to assess tasks in the classroom.

When I am preparing a task for my classroom, I first create a rubric (pictured above). I consider what the goals of the task are: are students generating ideas about how to solve a new type of problem? Are we solidifying understanding they should already have? Are they creating a new representation or algorithm? I do not always use all 4 pieces of the rubric, but instead use the ones that align most closely with my mathematical goals. I define each of these at a beginning, developing, accomplished, and exemplary level.

Conceptual understandings refer to the actual extent to which students understand the mathematics, as evidenced by their work and explanations. Justification and thinking I separate from this, as it refers more to the level of thinking presented in a students answer and in the discussion phase. When developing a rubric, consider the possibility a student is valiantly defending a wrong answer. While their conceptual knowledge might be low, their justification and thinking may still be high-level. I consider the terms and notations (equations, use of correct vocabulary) and the use of tools when applicable (ten frames, hundreds charts, fingers, etc.).

Once I have aligned the task to the mathematical goal, I am ready to think about how I will formatively assess to determine student progress. This is where the "formative five" come into play:

1. Observation: Usually a simple check list, this type of formative assessment allows teachers to quickly assess an entire class or large group. Using the rubric, pick a few things you can check off as you circle the room asking assessing and advancing questions. Prepare your rubric with student names and the few things you are looking for and you can quickly note a YES/NO during the lesson. This is particularly helpful if you want a snapshot of the group of students overall.

2. Interviews: Picking 2-3 key assessing and advancing questions, zero in on students you are seeking to understand better. Record their answers to each, calling back to the rubric to determine how they are developing understandings of the content.

3. Hinge questions: Posed at a key moment in the lesson, a hinge question is given to the whole class. A hinge question can determine whether or not the class is ready to move on, or needs additional time to explore the content at hand. It can be conceptual or procedural. Check these answers immediately and determine next steps.

4. Exit tickets: An old favorite, an exit ticket provides the opportunity for students to show their thinking on a similar problem or problem type that they explored during the task. Graded after the lesson, teachers then determine where the next lesson needs to go.

5. Show me: An obtrusive form of assessment that happens during the lesson. Stop a few students in their work, and ask them to show you their thinking physically, on the current or a different problem. Select these students intentionally.

The task cycle requires many opportunities for formative assessments throughout to determine where the learning goes next. Using rubrics and the formative five can help closely monitor student learning towards key mathematical goals!


Yours,

Ms. M

Welcome back! I am so glad you are joining in for session two, which will focus ideas for on launching mathematical tasks.

There are several goals when launching a high-quality task:

1. Eliminate all contextual and linguistic barriers for all students so they can focus on mathematics
2. Pose a problem
3. Generate interest
4. Activate background knowledge
5. Share logistical information: how long will students have to work? How will they be working (partners, individual, etc.)?

Hello! I am so happy you are joining me for session one of A New Vision for Mathematics today. Today will be a general overview of what the math task cycle is, its effect on student learning, and the three types of tasks.


One one hand, we have our old way of mathematics: introducing a concept and its vocabulary, teaching strategies, students completing work and then (maybe...) applying to a contextual problem to extend their learning. Imagine the motivation, ownership, and learning if this process was flipped on its head.

First, students develop ideas through a real-world problem or task. Engagement is high and so is cognitive demand. The teacher provides contextual information and tools, but no clear and tried path. No math word or strategy. Students work on their own and with a partner or group--they develop ideas all around the problem. Some students show misconceptions they have about number, some draw models, some develop diagrams, some use mathematics concepts they already know to solve. Answers are both wrong and write. As a class, a rich discussion is held. What happened? Which strategies worked? Which one's didn't? The teacher provides vocabulary and draws out key concepts and mistakes to be learned from. This is known as the "developing" phase.

Next, where does the learning go? Students have many ideas about their new concept, but have not been afforded time to use all the strategies. Teachers will share a targeted concepts--i.e., today, remember what Suzy Q. explained to us yesterday about how she used place value? Let's use that concept to see if it will help us on our task today. Another launch of an engaging task, more work with more focus on a strategy and concept, another rich discussion. Students are solidifying ideas about mathematics--this is the "solidifying" phase.

Finally, students have developed and solidified all important ideas about their mathematics topic. They are ready to work on their accuracy and fluency. The teacher launches a problem string, mathematics game, or task that requires repeated reasoning. The students work and re-work the tasks, focusing on using strategies flexibly and attending to the correct answers in problems. This is the "practicing" stage, and marks the end of the mathematics task cycle.

Because students have built the knowledge themselves, they are likely to take ownership and retain it. Because they built the knowledge conceptually FIRST, they are likely to apply it and have a deep understanding. And all throughout the cycle they have been working on engaging problems and confronting their own misconceptions.

The math task cycle is powerful when teachers use it in the classroom, and can be used to ensure all students have access to deep conceptual understanding. No more "I'm not a math person..." All students can be successful in mathematics through the learning cycle.

Join me next time for a deeper dive into planning for learning in the three stages!

Yours,


Ms. M

I love teaching comparing numbers and quantities every year in kindergarten! In kindergarten, these two standards address what kindergartners need to know and do to master number and comparisons: