*Hilary Agnello*

# Helping Students Analyze Their Own Mathematical Thinking

## Encouraging Deeper Mathematical Reflection

The traditional mathematics classroom of silent students in rows listening and observing the teacher work through examples as they take notes and practice similar problems on their own for homework is no longer considered to be an engaging or effective instructional practice. With the incorporation of the **Common Core State Standards**, a set of national content standards and mathematical practices that many states now use as the foundation of their math curriculums combined with the best practice, classrooms shift from the teacher being the ''sage on the stage'' to taking on the role of facilitator.

The classroom then fosters cooperative learning and encourages higher order thinking skills and requires teachers to develop effective questioning strategies beyond simply asking students for the answer to a skill-based question.

Effective math classrooms now include components of self-reflection and justified explanations for answers. To properly encourage this type of student development and thinking, teachers need to help students evaluate their own mathematical thinking and provide scaffolding that promotes deeper mathematical thinking.

Let's go over three strategies that teachers can build into their daily planning and instruction that can help create classrooms rich in mathematical discourse and that encourages students to evaluate and grow their own mathematical thinking.

### Plan For Opportunities

The first and easiest classroom strategy is to plan for opportunities during class for students to dialogue about math. This can be through having students explain how they got their answers to debating about the best strategy for solving a problem.

Creating opportunities for students to talk to each other about math allows them to compare in a structured way how well they can discuss and understand conversations about math as well as practice using math vocabulary.

### Ask Higher Level Questions

The second strategy is to intentionally ask math questions that go beyond the basic numeric answer. Sometimes teachers ask too many low-level computation based questions with the intention to use questioning as a classroom management strategy.

Asking a large amount of low-level skill questions in order to ''catch'' those students who are not paying attention devalues questions in general and promotes the idea that when the teacher asks a question it is to check for attentiveness instead of comprehension.

Asking higher order thinking questions can still have the same effect of encouraging students to pay attention, since they need to be prepared to answer a more challenging question. However, asking fewer questions that cause students to provide more than a single numeric answer allows students to process more thoroughly how they can provide an acceptable answer as well as evaluate whether or not they are prepared to answer questions of this degree.

Additionally, following up computation-based questions with some type of extension question such as ''how did you get that?'' or ''how do you know this answer is correct?'' has the same effect of encouraging student reflection on their own mathematical thinking.

### Bloom's Taxonomy

A third strategy incorporates Bloom's taxonomy, which is a highly useful tool for all teachers to incorporate into their planning and instruction practices. **Bloom's taxonomy** is a hierarchy of types of questions and activities that become progressively more challenging and higher order thinking the farther up the ladder (or sometimes visualized as a pyramid) one moves.

Starting at the top, the pyramid looks like this:

- Create
- Evaluate
- Analyze
- Apply
- Understand
- Remember

Bloom's taxonomy can be used as a visual aid in the classroom that students can use when working through their own mathematical problems. This can be a way that teachers encourage students to be very aware of their own ''standing'' in terms of what types of problems they are able to successfully answer.

However, an additional approach to using Bloom's taxonomy is for teachers to consistently incorporate at least one or two questions a day that ask students to think at the higher levels of Bloom's taxonomy and have students journal on these questions specifically.

Math journaling in general is not only an excellent instructional strategy to encourage higher order thinking skills, but is also a great way for students to actively reflect on their own mathematical thinking specifically as this thinking relates to higher order thinking questions.

## Lesson Summary

To encourage students to actively reflect and evaluate their own mathematical thinking, teachers need to incorporate a variety of questioning and instructional strategies. The **Common Core State Standards** is a set of national content standards and mathematical practices that many states now use as the foundation of their math curriculum, encouraging the teacher to take on the role of facilitator.

Three of these strategies can be adapted for any grade and classroom level:

- plan for opportunities - have students explain how they got their answers or debate about the best strategy for solving a problem.
- ask higher level questions - ask math questions that go beyond the basic numeric answer.
- incorporate Bloom's taxonomy - a hierarchy of types of questions and activities that become progressively more challenging and higher order thinking the farther up the ladder

Upon incorporating these strategies into their lesson plans and instruction, teachers will find that the mathematical discourse in their classrooms increases, and their students' awareness of their own mathematical thinking is more developed.

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Back*Helping Students Analyze Their Own Mathematical Thinking*

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