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--Resource: Getting started implementing writing in math.

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--Resource: Getting started implementing writing in math.

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In this article, the author provides steps that a teacher could take to implement students writing out their explanations in math class. It helps the teacher focus on what the desired outcome is before asking the students to simple write their explanations.

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I've always been an English-Language Arts kinda guy. Math was never my forte. When I learned that my new job would involve helping schools make the transition to the Common Core Standards for Math as well as English, I was nervous. How would I get up to speed on the math standards?

As I learned about the math standards, I saw connections to what I loved about the ELA standards and my anxiety diminished. The CCSS for math contain both grade level content standards that explain "the what" - what students should know and be able to do - and Standards for Mathematical Practice (SMPs) that explain "the how" - how students should learn and engage with one another in math class.

The SMPs apply to every grade level and are grounded in "important 'processes and proficiencies' that are highly valued in the field of math." Many of the things that I love about the ELA standards, such as the priority placed on evidence-based reasoning and the importance of explaining one's thinking, are present in the SMPs as well.

Make the Writing-Math Connection

When a school leader recently approached me about ways to incorporate a focus on Writing to Inform and Explain (CCSS ELA Writing Anchor Standard 2) in math class, I thought of the SMPs as a great way to make the connection. SMP 6 (attend to precision in communicating with others) is the standard that is the most natural fit for Writing to Inform and Explain, and also a great place to begin when introducing writing in math.
Excerpt from SMP 6 - Attend to Precision: "In the elementary grades, students give carefully formulated explanations to each other... Mathematically proficient students try to use clear definitions in discussion with others and in their own reasoning... by referring specifically to each important mathematical element, describing the relationships among them, and connecting their words clearly to those representations."
SMP 6 also states the importance of precision in terms of specifying units of measure, labeling the axes of a graph, and stating the meaning of mathematical symbols. To be clear, SMP 6 refers as much to oral explanations in small group discussions as it does to written explanations. Though the CCSS math standards don't emphasize writing, taking the additional step of having students write out explanations after discussing them is a great way to reinforce and assess what the SMP requires.

Connection to Writing to Inform and Explain in ELA

The ELA grade level standards for Writing to Inform and Explain also focus on precise language, clear definitions, and making the relationship among ideas explicit. Compared to the explanations students write in Reading or Writing class, written explanations in math are more of a combination of words and visual representations.

If departmentalized, math teachers should connect with their ELA counterparts to see what students already know about writing to explain, so both teachers can highlight the similarities and differences of writing explanations in ELA and math. If writing to explain is new to ELA teachers as well, this book from Jeffrey Wilhelm is a great place for middle and high school teachers to start.

Getting Started with Writing Explanations in Math

Here is a process I recommend for getting started with writing explanations in math, adapted from my own work coaching teachers. It is most powerful when used by a team of teachers who plan together.
Unpacking the Standard
1. Begin with the language of SMP 6 itself. First, read the "Commentary and Elaborations" document for SMP 6 (available for grades K-5 and 6-8) developed by Illustrative Mathematics. (You might be interested in checking out the recently published Teaching Channel video series sponsored by Illustrative Mathematics.) This document includes the language of the SMP, and helpful annotations and elaborations that explain what the standard looks like at different grade levels. As you read the standard, underline words and phrases you think are particularly important, and then discuss with a colleague what you underlined and why.

2. Next, make a simple T-Chart with elements of SMP 6 that students can do now, and things that are new or are a challenge for students.
Seeing the Standard in Practice
3. After you have an idea of what the standard asks of students, watch videos of students explaining their thinking to get a better idea of how it looks in an actual classroom. Inside Mathematics has a series of several videos of SMP 6 in action at different grade levels. Use insights from these videos to refine your list from Step 2.

4. Now, brainstorm what a good written explanation in math should contain. Return to your T-Chart from Step 2 and determine how these elements would show up in student writing, keeping in mind it may be a combination of words, graphs, equations, and drawings. These "look fors," based on SMP 6 and developed for a school district in Michigan, are a great starting place. Make a list of the characteristics of an effective written explanation in math for your grade level and create an anchor chart. Then identify characteristic(s) you want to target first.

5. Examine your unit plan for an upcoming unit, and determine content standards that would be a good fit for SMP 6 and writing explanations. Craft an objective targeting a specific characteristic (Step 4) you want to address. Then work as a team to create or adapt existing models of a good math explanation highlighting this characteristic. Share these models with students by doing a think aloud as you write your explanation, to help them see the connection between writing and thinking. Once students get comfortable writing explanations, have them examine examples written by their peers. Make a math explanation rubric as a class and post it so everyone can see!

Assessing Student Explanations

If students are going to be expected to explain their thinking, it helps to have a rigorous task that asks them to do so. The team at Mathematics Assessment Project has gathered a number of assessments that align with SMP 6. Requiring a written explanation of the steps or procedure students took in completing one of these assessments is yet another way to incorporate Writing to Explain in your instruction.

Time to Write!

Asking students to write out an explanation of how they solved a problem can give you unique insight into student thinking that goes much further than "show your work" ever will. After students can write explanations, switch the focus to making arguments about which approach to solving a problem is most effective ( SMP 3). It pairs nicely with Writing Anchor Standard 1, but that's a topic for another blog.

Share your ideas below about how you use or plan to use writing in math.
Ryan McCarty is a coach with Achievement Network, a nonprofit organization that helps school leaders support teaching that is grounded in standards, data, and the best practices of schools across the country. He partners with schools in Massachusetts. Prior to that, he was a literacy coordinator, instructional coach, and teacher in Chicago, IL. All views are his own. Follow Ryan on Twitter: @RyanP_McCarty.

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I've always been an English-Language Arts kinda guy. Math was never my forte. When I learned that my new job would involve helping schools make the transition to the Common Core Standards for Math as well as English, I was nervous. How would I get up to speed on the math standards?

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Give the students sentence starters for their explanations or reasonings.

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--Resource: writing across the curriculum(Just ideas) 

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--Resource: writing across the curriculum(Just ideas) 

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This article provides different strategies that can be implemented in the classroom to promote writing in mathematics. I could see myself using some of these strategies in my classroom to help not only the students write about their thought process, but it would help me as a teacher see their thought process.

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--Resource: How to teach conceptual understanding.

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--Resource: How to teach conceptual understanding.

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This author discusses five tips that have helped her develop conceptual understanding for math in her classroom. It also includes a video about how one teacher teaches for understanding. She focuses on a simple question first and then build on the knowledge from there.

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As a teacher of mathematics who focuses on inquiry-based instruction, I often get asked how I go about developing my students' conceptual understanding. Early in my teaching career, I became very good at teaching procedures through direct instruction.

As my teaching methods have evolved, especially since embedding the math practices in every lesson, I've pushed myself to use inquiry-based tasks and hands-on manipulatives with my students as a way to help them develop their conceptual understanding. Yet there are still topics (such as solving inequalities with a negative coefficient) where I feel unprepared to teach in this way.

I recently got a sneak peak of Leah Alcala's new video, Concept First, Notation Last. While Leah is probably best known on Teaching Channel for her My Favorite No and Highlighting Mistakes strategy videos, this new example of practice is a chance for us to see how Leah helps her students develop conceptual understanding in math. After watching, I felt as if some of my own questions on the topic had been answered.

Here are my 5 takeaways, which are now reminders to myself and tips for you:

1. Belief
Leah Alcala has the utmost belief that her students will be able to access and attempt the task at hand. In the video, you'll hear her make the statement, "Everyone likes to learn." It's clear that her beliefs move students forward. Take inventory of yourself. Do you believe that ALL students in your classroom can access mathematical ideas without being told what to do and how to do it? To continue to deepen your beliefs, I recommend reading Jo Boaler's research on how all students are capable of developing mathematical ideas.

2. Sense Making
When my teaching started to evolve into one that focused more on conceptual understanding rather than rote memorization, I started asking one question of all my students: "How/Why does your solution make sense to you?" I found that by developing justification as a norm, I could understand exactly what students believed, why they believed it, and respond with more effective questions. Moreover, in order to have those conversations, the task they were responding to had to be a rich, inquiry-based task. That meant I was no longer teaching tricks or math shortcuts. Students had to be able to construct viable arguments and critique the reasoning of others. In the video, you'll see Leah posing carefully developed tasks alongside constant questioning, as opposed to teaching mathematical tricks. This allows her students to use sense making to come to a conclusion on their own.

3. Scaffolding
From the outset of this lesson, Leah keeps the numbers easy to manipulate while the mathematics remains demanding. If the numbers become too difficult at the outset, students can get overwhelmed and will often shut down. As Leah mentions, easier numbers help give an access point to all students. Over time, you can increase to more challenging numbers while maintaining challenging thinking as well.

4. Time
Conceptual ideas are not built in one day or even two. They are developed after repeated exposure to a particular mathematical idea in various contexts. Students have to struggle and resolve that struggle to internalize a concept. This process takes time. Leah mentions that the one day her class is being filmed will, by itself, not be sufficient for complete learning to occur. In my classroom, this means a learning objective may remain consistent for a week or two as we are developing the idea.

5. Multiple Representations
Leah allows her students to engage with the mathematical idea of solving inequalities through graphs, lists, and/or mathematical notation. When developing conceptual understanding, it's imperative to give students freedom of choice in how they might potentially respond. Narrowing to one representation too often causes students to try to find the one correct path towards a solution, rather than thinking expansively and for themselves.

[video_embed src=" https://www.teachingchannel.org/videos/teaching-inequalities/embed.js" width="480"]

Since I've begun to focus on the development of conceptual understanding, I've run into some challenges. Students and parents alike want me to teach short cuts and algorithms. Yet, when I hear students talking and thinking mathematically, I'm certain that this struggle will prepare them to be risk-takers, not only in my classroom but in their daily lives, too. My hope is that this will lead to self confidence and an internal belief that when faced with any problem, they have ideas worth exploring.

I encourage you to watch this video and think about how developing conceptual understanding is both doable and essential in your classroom. I would love to read your responses in the comments section below on how you were able to take these ideas and apply them with your students!
Crystal Morey is a middle school math teacher in Enumclaw, WA. She has served on the county's Common Core transition team, and is a teacher leader on a statewide project with Washington State's Office of the Superintendent of Public Instruction and the Mathematics Education Collaborative.

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5 Tips To Help Develop Conceptual Understanding In Math

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As a teacher of mathematics who focuses on inquiry-based instruction, I often get asked how I go about developing my students' conceptual understanding. Early in my teaching career, I became very good at teaching procedures through direct instruction.

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This resources touches on many different parts that are essential to teaching math.  I found the most information for my topic in section two.  It gave clear descriptions of what actions students should be able to perform in a math classroom.

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--Resource: What is conceptual understanding 

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--Resource: What is conceptual understanding 

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This article discusses conceptual understanding in mathematics.  There is more to conceptual understanding than just saying how you solved a problem.  It is being able to apply the knowledge in other situations.

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In this article it discusses how writing in math deepens a students understanding of the concepts.  It also discusses how writing in math helps develop reasoning skills.

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--Resource: "Teaching for Understanding with Technology"

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--Resource: "Teaching for Understanding with Technology"

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This book explains what it means to teach for understanding.  It also discusses how to incorporate technology in order to teach for understanding.

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--Resource: "Problem with Math Is English : A Language-Focused Approach to Helping All Students Develop a Deeper Understanding of Mathematics"

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--Resource: "Problem with Math Is English : A Language-Focused Approach to Helping All Students Develop a Deeper Understanding of Mathematics"

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This book focuses on how math has a language of itself.  Having to learn not only the skills but the language of math as well is a struggle for many students.

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--Resource: "So You Have to Teach Math?"

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--Resource: "So You Have to Teach Math?"

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In this text I focused on chapter seven which discussed how to incorporate writing into math.  It discusses how being able to explain their thinking verbally must come before writing down an explanation. (I got this resource from Marygrove library, you may need to log in to see the whole book)

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Implementing a math journal for students to write down ideas may be beneficial.

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--Teaching math for understanding strategy.

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--Teaching math for understanding strategy.

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This video gives an example of how one teacher teaches math for understanding.  This lesson is student led with the teacher assisting by asking questions to continue their thinking.  The students are the ones coming up with different explanations of how they got a particular answer.

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Being able to apply what they have learned to other situations.

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Complete understanding includes not just solving a problem and understanding how to solve it, but also taking that knowledge and using it in another context

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Having students learn math through discovery on their own. Have them think about what they believe the reasoning is first.

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This article discusses how students feel about math in general.  It also discusses how teachers can make math a subject students can learn and increase their ability in as well as being strictly performance based.

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For many students, math class is the subject of nightmares. Many who enthusiastically show off their counting skills in pre-school years later consider math class the most loathed hour of the school day. The problem, according to a Stanford education professor, is a matter of mindset.

"All children are different in their mathematical thinking, strengths and interests," said Jo Boaler, a professor of mathematics education at Stanford Graduate School of Education.

Boaler has studied how students learn math, to identify why so many students end up hating or fearing math, and, too often, underachieve. She has followed thousands of students through middle and high schools to ascertain exactly how they learn and what are the most effective ways they reach their potential.

In her new book, Mathematical Mindsets, Boaler describes how society has traditionally valued one type of math learner - one who can memorize well and calculate fast - as opposed to others who have the same potential but may be deeper, slower and more creative. The book offers research evidence and resources for teachers and parents to help children learn math well, in the classroom and at home.

Test-taking pressures

The "testing culture" in American schools is a major problem for math learners, Boaler said, noting that exams and tests are far more pervasive in math than other subjects.

"Students from early ages realize that math is quite different than other subjects, and that it depends heavily on taking tests and answering uninteresting questions."

This combination of frequent testing, procedural teaching and uninteresting questions disconnected from students' lives causes many students to regard math as a "cold, lifeless subject" without a human connection, she said.

Many students, she said, believe incorrectly that to be good at math they need to be fast at math - too many classes emphasize speed in math and memorization.

Boaler has published research and strategies on the best ways for students to learn math "facts" without the fear and anxiety that many students feel when given timed tests and flash cards.

For example, speed pressure in math causes the early onset of math anxiety for many students, she said. And children who think more slowly and deeply are often turned off by math because they see it as a subject of shallow recall of facts and rules.

Opening up mathematics

The good news is that any student's negative perception of math can dramatically change, Boaler said. In her recent summer math camp for seventh- and eighth-grade students, which can be seen in this video, students learned to enjoy math and they improved their achievement dramatically through open, creative math lessons.

Boaler works with Stanford psychology Professor Carol Dweck to deliver growth mindset interventions to teachers and students. A "growth mindset" is built around the idea that most basic abilities can be developed and expanded upon through dedication and hard work. By contrast, with a "fixed mindset," people believe their basic qualities, like their intelligence or talent, are simply fixed traits.

"When we open up mathematics and teach broad, visual, creative math, then we teach math as a learning subject instead of as a performance subject," Boaler said.

Boaler offers resources on how to develop growth mindsets for math learners. Her new Stanford center, youcubed, which is funded by the Bill & Melinda Gates Foundation, provides free mindset math lessons that were used in over 100,000 schools last fall. The website has attracted millions of visitors in the last month alone. She also has online courses for teachers and parents and for students to help them teach and learn math.

When schools teach young people growth mindset mathematics, Boaler said, the outcome is that the subject becomes deeper and filled with more connections, so students enjoy it more and achieve at higher levels.

Math for today's tech world

Boaler has also worked with the Program for International Student Assessment tests team, analyzing their data set of 13 million students worldwide. They found that the lowest achieving students are those who use a memorization strategy. The highest performers are those who see math as a set of connected, big ideas.

Boaler agrees with high-tech leaders such as Conrad Wolfram, who supports "broad, multidimensional" mathematics, which involves problem solving, modeling, reasoning, making connections and thinking deeply.

She points out that students entering the 21st-century workplace need to be able to ask questions of data, map out mathematical pathways and reason quantitatively - routine calculations, after all, will be performed by machines. She cites research studies that demonstrate how such an approach can engage more young people and put them on a path to success.

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Research shows how children can enjoy and succeed in math, Stanford expert says

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For many students, math class is the subject of nightmares. Many who enthusiastically show off their counting skills in pre-school years later consider math class the most loathed hour of the school day. The problem, according to a Stanford education professor, is a matter of mindset.

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This article discusses a strategy implemented in a school called Comprehensive Mathematics Instruction(CMI). Focusing on students getting to know the "how" and "why" of a math problem.

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- short answer explanations
- true/false questions with explanations
- multiple choice of an explanation

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Does complete understanding affect a students perception of math?