The past semester has been the busiest season of my life as I became a part of my dream school. I am a mathematics coach at a STEM high school in northeast Ohio, where I am a mathematics coach who guides and inspires all of the freshmen in their journeys in algebra 1, geometry, and algebra 2. I have the opportunity to collaborate with the best group of people I have ever worked with, and we have the privilege of challenging students with rich problems that are relevant to our community.

# Math Explorers

## Friday, January 2, 2015

## Sunday, July 7, 2013

### My Vision

# My Vision:

To make

*all*students participators in the math classroom.**What does it mean to make students participators in the mathematics classroom?**- To make students participators in the mathematics classroom means that students are actively engaged in the learning process, taking part in instructional activities instead of exclusively listening to a lecture.

**How do students become participators in the mathematics classroom?**

- Through
**communication**: every student should learn to communicate about mathematics. This can happen through classroom discussions and small group conversations. Technology can assist in communication through texting and social networking sites such as Edmodo. - Through
**exploration:**students gain a much deeper understanding of mathematics if they are given opportunities to discover mathematical concepts for themselves. This can be achieved through technological applications that let students visualize mathematical concepts, through inquiry based-learning projects where students are asked to investigate a mathematical question or problem, or any other means of student-driven discovery. - Through
**critical thinking and reasoning:**students should analyze mathematics by recognizing patterns that they observe or discover and logically developing mathematical arguments based on their observations. These reasoning skills are a critical part of the mathematical skills they will need for their future careers. - Through
**collaboration:**cooperative learning activities enhance all of the above activities, by allowing students to learn from each other and sharpen each other’s ideas.

**Why should students become participators in the mathematics classroom?**

- Participation increases
**student motivation**. Students are more interested in mathematics when they are involved in the learning process! - Participation
**engages**students in the classroom. When students are exploring math through hands-on representations or technological tools, or when they are reasoning about math and discussing ideas with their peers and teacher, they must use all of their senses in the learning process. This makes mathematics more concrete and easier to understand. - Participation provides real opportunities for
**differentiation.**When students are exploring math, or developing mathematical arguments, a teacher can provide different levels of exploration and critical thinking for students who are at different academic stages. More advanced students can be driven to dig deeper than other students. The teacher can provide scaffolding interventions for students who need them. **The Common Core Standards demand student participation in the math classroom**. Mathematics Practice #3 requires students to “construct viable arguments” (*critical thinking*) and “critique the reasoning of others” (*communication*and*collaboration*). Practice #4 requires students to “model with mathematics” (*explore*real world scenarios with math). Students are also required to “make sense of problems” (#1), “reason abstractly and quantitatively” (#2), and “look for and express regularity in repeated reasoning” (#8), which require critical thinking and are enhanced by exploration, communication, and collaboration.

## Tuesday, May 14, 2013

### Algebra: A Mathematical Poem

**Algebra**

When I met the unknown

I stopped,

And cried, familiar only

With the metronomic

Repetition of numbers.

What was this letter doing

In the midst of my math book?

From one moment to the next,

Life went from simple

To impossible,

When I could no longer

Plug questions in a calculator

To get an answer.

But when I started treating the unknown

As a number

(Shaking it like a gift

That I might find a clue),

The pieces fell in place,

And I found the answers

Right beneath my nose.

## Thursday, May 9, 2013

### 2013 NCTM Conference: "Exploring the Common Core Practices in Secondary Classrroms"

2013 NCTM Conference (4/19/13 - 3:30 pm - Kristen Bieda and Samuel Otten):

My intention in this session was to learn more about the Common Core Mathematical Practices. Kristen and Samuel provided details on these practices embedded in the standards, but even more importantly, they suggested ways of implementing the standards in a high school mathematics classroom.

The Practices are defined as follows (the links were not provided in the session):

Standard 1: Make sense of problems and persevere in solving them
Standard 2: Reason abstractly and quantitatively
Standard 3: Construct viable arguments and critique the reasoning of others
Standard 4: Model with mathematics
Standard 5: Use appropriate tools strategically
Standard 6: Attend to precision
Standard 7: Look for and make use of structure
Standard 8: Look for and express regularity in repeated reasoning

Kristen and Samuel noted that these practices require students to "do" mathematics rather than just following mathematical procedures laid out by the teacher. They suggested that "doing" mathematics requires the following:

Kristen and Samuel promoted the importance of modifying tasks to promote mathematical reasoning. They provided several ideas for doing so:

My intention in this session was to learn more about the Common Core Mathematical Practices. Kristen and Samuel provided details on these practices embedded in the standards, but even more importantly, they suggested ways of implementing the standards in a high school mathematics classroom.

The Practices are defined as follows (the links were not provided in the session):

Kristen and Samuel noted that these practices require students to "do" mathematics rather than just following mathematical procedures laid out by the teacher. They suggested that "doing" mathematics requires the following:

- Students participate in complex and non-algorithmic thinking.
- Students explore mathematical concepts.
- Students self-monitor their problem-solving processes.
- Students access relevent knowledge.
- Students examine task restraints.
- Often, students will experience a level of anxiety as they wrestle with mathematics in a more real way.

Kristen and Samuel promoted the importance of modifying tasks to promote mathematical reasoning. They provided several ideas for doing so:

- Ask students what happens in the nth case of a pattern.
- Ask students to justify their conjectures.
- Reduce scaffolding, so that students are prompted to seek out strategies for problem-solving.
- Ask students to identify prior knowledge that they assume is true and justify the use of this knowledge.
- Ask students to define key mathematical objects.
- Ask students to model a problem through visual representations.

## Wednesday, May 8, 2013

### 2013 NCTM Conference: "Take Time to Question the Questions"

2013 NCTM Conference (4/19/13 - 12:30 pm - Mark Howell):

I've come to believe that asking the right questions is one of the most important parts of building student understanding of new mathematical concepts. One of my goals in coming to the NCTM conference was wanting to continue to sharpen my questioning skills. I was therefore eager to learn from Mark's session about "questioning the questions".

Mark started the session with a bold statement: he suggested that as beneficial as multiple representations of mathematical concepts are, the use of multiple representations is not sufficient. He argued that linking between and among the representations is where learning takes place, when student understand how the representations are connected together.

Mark proposed that good questions are those that come from powerful connections between representations. These questions prompt students to explore these connections and examine what the representations actually mean.

Mark noted that it is possible to ask a lot of questions in the classroom without building a deeper understanding of mathematics. He noted that we've all seen procedural questions that ask students to provide the next step in a familiar mathematical routine. These questions tend to be very repetitive and removed from context. But students can memorize procedures without knowing how to think about math or understanding what the procedures really mean.

Teachers often provide shortcuts for students that allow them to solve problems without thinking about how math really works. This can be detrimental for students in the long run. For example, students may have memorized the "vertical line test" without knowing the properties of a function.

Mark noted that questions are essential for revealing deeper mathematical realities. But it is difficult to craft good questions that prod students into these deeper realms. There are four different levels of questions:

1. Questions that elicit feedback from students.

2. Questions that ask for factual responses.

3. Questions that ask students to explain a procedure.

4: Questions that ask a student to explore math, make conjectures, explain and verify their results, or talk about connections between representations.

Most teachers' questions remain in the first three levels, but these questions are all very shallow compared to the fourth level of questions. It is important that we consider how to craft questions that take students to this deeper level.

I've come to believe that asking the right questions is one of the most important parts of building student understanding of new mathematical concepts. One of my goals in coming to the NCTM conference was wanting to continue to sharpen my questioning skills. I was therefore eager to learn from Mark's session about "questioning the questions".

Mark started the session with a bold statement: he suggested that as beneficial as multiple representations of mathematical concepts are, the use of multiple representations is not sufficient. He argued that linking between and among the representations is where learning takes place, when student understand how the representations are connected together.

Mark proposed that good questions are those that come from powerful connections between representations. These questions prompt students to explore these connections and examine what the representations actually mean.

Mark noted that it is possible to ask a lot of questions in the classroom without building a deeper understanding of mathematics. He noted that we've all seen procedural questions that ask students to provide the next step in a familiar mathematical routine. These questions tend to be very repetitive and removed from context. But students can memorize procedures without knowing how to think about math or understanding what the procedures really mean.

Teachers often provide shortcuts for students that allow them to solve problems without thinking about how math really works. This can be detrimental for students in the long run. For example, students may have memorized the "vertical line test" without knowing the properties of a function.

Mark noted that questions are essential for revealing deeper mathematical realities. But it is difficult to craft good questions that prod students into these deeper realms. There are four different levels of questions:

1. Questions that elicit feedback from students.

2. Questions that ask for factual responses.

3. Questions that ask students to explain a procedure.

4: Questions that ask a student to explore math, make conjectures, explain and verify their results, or talk about connections between representations.

Most teachers' questions remain in the first three levels, but these questions are all very shallow compared to the fourth level of questions. It is important that we consider how to craft questions that take students to this deeper level.

## Tuesday, April 30, 2013

### 2013 NCTM Conference: "Technology: A Portal to Exploration and Discovery"

2013 NCTM Conference (4/19/13 - 8:00 am - Kenn Pendleton):

A big theme through the NCTM conference was using questions to prompt students to explore mathematics. I am a big proponent of inquiry-based learning, so I was excited to find other teachers with similar passions. I was able to collect a lot of fresh ideas.

Kenn suggested that we can use technology as a tool to enable students to explore mathematics. He suggested that we as teachers should ask questions which require students to use technological tools (such as graphing calculators) to explore and answer.

Kenn strongly encouraged the use of technology to allow students to explore the connection between graphs and their symbolic representations. This connection is very important for helping students to visualize math and to build a deeper understanding of what mathematical symbols actually represent.

Kenn encouraged teachers to challenge students to create conjectures based on their observations using technology. They can then test these conjectures by contrasting different scenarios and evaluating whether their conjectures are true or not. This can given them a fuller understanding of mathematical principles than the teacher given a list of facts that go in one ear and out the other.

Technology can be a powerful tool for comparing the graphs of different functions. This can help students contrast functions with their inverse, tranlations, and other properties of functions.

A big theme through the NCTM conference was using questions to prompt students to explore mathematics. I am a big proponent of inquiry-based learning, so I was excited to find other teachers with similar passions. I was able to collect a lot of fresh ideas.

Kenn suggested that we can use technology as a tool to enable students to explore mathematics. He suggested that we as teachers should ask questions which require students to use technological tools (such as graphing calculators) to explore and answer.

Kenn strongly encouraged the use of technology to allow students to explore the connection between graphs and their symbolic representations. This connection is very important for helping students to visualize math and to build a deeper understanding of what mathematical symbols actually represent.

Kenn encouraged teachers to challenge students to create conjectures based on their observations using technology. They can then test these conjectures by contrasting different scenarios and evaluating whether their conjectures are true or not. This can given them a fuller understanding of mathematical principles than the teacher given a list of facts that go in one ear and out the other.

Technology can be a powerful tool for comparing the graphs of different functions. This can help students contrast functions with their inverse, tranlations, and other properties of functions.

## Thursday, April 25, 2013

### 2013 NCTM Conference: "Discover New Ways to Make High School Math Meaningful"

2013 NCTM Conference (4/18/13 - 2:30 pm - Tim Pope):

Tim Pope is the curriculum manager at Kendall Hunt Publishing Company. While this made his session a little more of an advertising campaign, he had some good ideas to share as well.

One statement Tim made really stuck with me, perhaps more than any other at the conference. He said, "If we can remind students of a story that kids can remember, we'll always have something to build on." He elaborated on this idea further, suggesting that if we make math real to students and get them involved in the classroom, they'll remember it much better than they would if they memorized a bunch of formulas. They'll remember even better if a teacher makes math fun as well, or allows students to get up and move around, physically participating as well as mentally participating.

Tim got the audience to participate in an activity identical to one he would use with students. We calculated the amount of time it takes for a specified number of people to complete the "wave", and plotted this data in a scatterplot. We then used this data to create a line of best fit and discussed the properties of this line.

Tim advocated the Van Heile structure for building an understanding of mathematical concepts, and gave an example of each.

Level 0: Visual - students are able to visually identify shapes, patterns, etc., but can't explain why. "It is a rectangle because it looks like a door."

Level 1: Descriptive - students are able to describe the properties of shapes and patterns, but are unfamiliar with the ways different properties interact with each other. "It is a rectangle because it is a quadrilateral with four right angles."

Level 2: Abstract - students can more fully describe the ways the properties of shapes and patterns interact with each other and understand what makes each individual group unique. "I know it’s a rectangle if it’s a parallelogram with four right angles."

Level 3: Proof - students can prove why the properties of shapes and patterns are true. "I can prove it is a rectangle if it’s a parallelogram with one right angle."

Level 4: Rigor - students understand a concept so fully they don't need to consider the basic properties of the concept. "I know that is a rectangle - what would happen if I tried to put it on a sphere?"

Tim suggested that we should challenge students to wrestle with higher-level mathematical questions, that they may gain a fuller understanding of the mathematical concepts involved.

Tim Pope is the curriculum manager at Kendall Hunt Publishing Company. While this made his session a little more of an advertising campaign, he had some good ideas to share as well.

One statement Tim made really stuck with me, perhaps more than any other at the conference. He said, "If we can remind students of a story that kids can remember, we'll always have something to build on." He elaborated on this idea further, suggesting that if we make math real to students and get them involved in the classroom, they'll remember it much better than they would if they memorized a bunch of formulas. They'll remember even better if a teacher makes math fun as well, or allows students to get up and move around, physically participating as well as mentally participating.

Tim got the audience to participate in an activity identical to one he would use with students. We calculated the amount of time it takes for a specified number of people to complete the "wave", and plotted this data in a scatterplot. We then used this data to create a line of best fit and discussed the properties of this line.

Tim advocated the Van Heile structure for building an understanding of mathematical concepts, and gave an example of each.

Level 0: Visual - students are able to visually identify shapes, patterns, etc., but can't explain why. "It is a rectangle because it looks like a door."

Level 1: Descriptive - students are able to describe the properties of shapes and patterns, but are unfamiliar with the ways different properties interact with each other. "It is a rectangle because it is a quadrilateral with four right angles."

Level 2: Abstract - students can more fully describe the ways the properties of shapes and patterns interact with each other and understand what makes each individual group unique. "I know it’s a rectangle if it’s a parallelogram with four right angles."

Level 3: Proof - students can prove why the properties of shapes and patterns are true. "I can prove it is a rectangle if it’s a parallelogram with one right angle."

Level 4: Rigor - students understand a concept so fully they don't need to consider the basic properties of the concept. "I know that is a rectangle - what would happen if I tried to put it on a sphere?"

Tim suggested that we should challenge students to wrestle with higher-level mathematical questions, that they may gain a fuller understanding of the mathematical concepts involved.

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