developing a classroom culture that supports a problem solving approach to mathematics

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Twelve Ways to Make Math More Culturally Responsive

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Especially in light of the filmed police shootings of African Americans this year, more attention is being paid by educators toward culturally responsive teaching. It might be a bit more obvious about how to apply that concept in English and history classes than in math, but it’s also very possible—and necessary—to do the same in that subject.

Today’s guest contributors are Chiquita Jenkins, Autumn Kelley, James Ewing, and Cindy Garcia. Chiquita, Autumn, and James were guests on my 10-minute BAM! Radio Show . You can also find a list of, and links to, previous shows here.

Community, connections, & collaboration

Chiquita Jenkins teaches 2nd grade in New York. She is a doctoral candidate at St John’s University and will be graduating in May 2021 with a Ph.D. in educational literacy:

In order to reach learners, creating instruction that is comparative and applicable to students’ experiences can be a treasured tool. Students may be captivated by an understanding that math exists all around us. From the arrangement of rows and columns seen in a neatly planted rose garden to algebraic formulas used to build sturdy homes, bridging the gap between math and the real world can introduce students to learning in a more culturally responsive way. Culturally responsive instruction as defined by Geneva Gay as “using cultural knowledge, prior experiences, frames of reference, and performance styles of ethnically diverse students to make learning encounters more relevant to and effective for them.” Components of culturally responsive instruction can be addressed within math practices when the following steps are put into place:

Building Community

The most important aspect of being a culturally responsive educator is building relationships with culturally and linguistically diverse students. Instead of viewing variances amongst students as shortcomings, educators can foster relationships with students in order to achieve a culturally sustaining instruction. Strategies such as talking with students in both structured and nonstructured conversations and talking with students’ parents can increase students being placed at the center of learning to increase engagement in ways that are culturally responsive.

Real-World Connections

Putting acquired math skills to use in real-world situations is a great way to help students connect diverse culturally experiences and become excited about math. Designing a standards- based math lesson that connects students’ language and cultural aspects of students’ background can foster mathematical identity development. For example, students can role play a real-world math scenario in which a peer is trying to determine the better deal for traveling to school and home for a week—paying the daily fare of $2.75 or buying a weekly pass for $33.00. Incorporating real-world connections into math practices in a culturally responsive manner can elect responses from students in ways that incorporate diverse ways of thinking, learning, and communicating.

Student Collaboration

Research affirms that students’ achievement drastically improves when students have opportunities to collaborate (Kagan 2010). Cooperative learning demonstrates the positive effects of interdependence while highlighting the importance of personal accountability amongst students. When students respond to small groups or in pairs, they may experience far less pressure because their responses are more private. For example, turn and talks allow for variations in how students from different cultural groups desire to communicate. In math application, students acquiring English can receive assistance with interpreting and understanding math equations from their partner while also eliciting language and communication.

If teachers want to be culturally responsive, they must invest the time to study their culturally and linguistically diverse students. When planning for math instruction, teachers can increase its culturally responsiveness in ways that can connect students to their lives and experiences inside and outside the classroom.

Gay. G. (2010). Culturally responsive teaching: Theory, research, and practice (2 nd ed.) New York, NY: Teachers College Press.

Kagan, S. (2010) Excellence & equity . Kagan Online Magazine. Retrieved from www.kaganonline.com/free_articles/dr_spencer_kagn/266/Excellence-amp-Equity

“Make it social”

A Washingtonian, a happy member of the D.C. public school system, with graduate degrees from Harvard School of Education and Johns Hopkins University, Autumn Kelly’s digital island of resources can be found here :

The goal of every teacher is to reach every child. Instructional techniques that are culturally responsive support teachers as they build math achievement in diverse classroom settings. Here are four ideas to make K-12 math instruction more effective among culturally diverse math groups.

Build Bilingual Communication Into Presentations of Math Instruction

Teachers can present elements of a math problem—street signs, grocery list, or directions within a recipe—in students’ native language. Linguistically familiar presentations of math content will build interest. Students are often willing to channel effort to address a challenge to which they can relate.

Language familiarity for students who speak English but use vernacular that is different from traditional print can also be reached with these methods. Presenting word problems in the form of text messages between two people would allow a teacher to incorporate the use of slang or popular regional terminology in the content of the math problem. The use of culturally sensitive language terms can be used to reach students from diverse backgrounds.

Make It Social—Create Math Experiences That Connect With Others

Students from culturally diverse backgrounds relate well to experiences that involve making connections with others. Have students work in pairs to either solve math problems or create math problems. As the teacher, when you match the pairs, open the work time with a quick “get-to-know you” activity to really build a personal bond between students in the math groups.

Embed the Local Community Into Your Math Instruction

Students who are culturally grounded are often deeply connected to the community they live in. Community connections can be a powerful way to integrate math into a student’s world as well as build personal investment in the student’s ability to use math to solve social problems.

For younger students, teachers might have students look at photographs of their neighborhood (grocery store, houses, mailbox areas {within apartment buildings} and inside of local stores) for price marking or signage and other community areas that feature numbers. Students can then use these photos to identify numbers and relate the message or meaning of the numbers to the familiar areas around them. This technique empowers young, culturally diverse learners to see the abstract nature of math as an integral part of their daily lives.

Older students can use community elements such as the local carryout menu and current or local ads from a car dealership. Students can apply math concepts to problem solve local math questions. Examples of local math problems in student communities could include designing a local playground or redesigning a housing unit. These kinds of activities let students apply math formulas to problem solving.

Oral-Based Instruction for Math Concepts

Culturally diverse students often come from a rich heritage of oral communication. Over generations, many cultures have used the power of the spoken word to share knowledge and preserve important ideas.

Math instruction in the classroom setting can be channeled into the traditions of oral communication to make it culturally relevant to students from diverse backgrounds. Using memory strategies such as mnemonics to learn formulas or key concepts, connecting math strategies to rhythm and music, partnering the recitation of math facts to movement/dance, and incorporating games which require active oral processing for participation are powerful ways to use the tradition of oral language as a path to mastering math content.

Experiences of community, traditions of oral language and dance, and incorporating elements of local and native language are all ways to deepen the connection between math instruction and culturally diverse students.

Role models, manipulatives, and literature

Dr. Jim Ewing is an assistant professor and author of the book Math for ELLs, As Easy as Uno, Dos, Tres. Jim provides motivating, relevant, and strategy-driven workshops for teachers that get results. To learn more about his workshops and book, go to EwingLearning.com:

First, let’s briefly discuss why it is so important to be culturally responsive in math. In Jo Boaler’s bestseller, Mathematical Mindsets , she explains how each and every student should have the belief that they can do math. However, if educators are not culturally responsive, some students may develop the belief that math is for “others,” not for them. Bottom-line: If we are culturally responsive, each student will feel like they can be successful.

Being culturally responsive is making students feel valued by celebrating their culture. We need to get to know our students, but we have to be careful of stereotypes. For example, when I was a child, our family moved to Wales for a year. Many of my new friends asked me how many TVs I had in my house. They assumed that since I was an “American,” then I would have many TVs—in fact I grew up without a TV in my house. In other words, while it is imperative for educators to learn about students’ customs and cultures, we have to be careful with assumptions. With this in mind, here are three strategies that educators may consider when teaching math.

Select relevant role models. Take 30 seconds and think of as many superheroes as you can. How many of those superheroes are not white men? The point is that we need to go out of our way to offer students role models that look like them. We should examine the posters we have in our classrooms and the videos we might show. Are we showing our students examples of Latinos doing math successfully? For example, the Aztecs were good at math and even used fractions centuries ago. They could be positive role models for our students. For motivation before a lesson, educators might say, “Let’s pretend we are Aztec mathematicians while we solve these fractions.” By being more inclusive, we can empower our students

Use culturally responsive manipulatives. Most educators know the value of using manipulatives, but have we ever considered what kind of manipulatives we use? For example, instead of just randomly using multiple colored counters in math class, we could purposefully celebrate our students’ cultures. We can make Mexican American students feel special by using red, white, and green counters and point out that those are the colors of the Mexican flag. Another way to celebrate Mexican culture in math is using “frijoles” for counters.

Incorporate culturally responsive literature. It is becoming common practice to engage students in mathematics by reading books to students at the beginning of the lesson. This is beneficial because it develops students’ academic language, which is imperative in all subjects, including math. However, we also need to consider what books we are reading to our students. The books we choose must be culturally responsive and “talk to” the experiences and backgrounds of our students. I have seen many books in classrooms translated into Spanish with storylines about white, middle-class children. The Spanish-speaking students may appreciate hearing Spanish, but the books should also be about the students’ cultures. Being culturally responsive should be an integral part of our curriculum and done on a daily basis.

Educators, try using the strategies above. You might find they are as easy as “uno, dos, tres.”

Including challenging and complex work

Cindy Garcia has been a bilingual educator for 15 years and is currently a districtwide specialist for PK-6 bilingual/ESL mathematics. She is active onTwitter @CindyGarciaTX and on her blog :

One of the main ways to make mathematics more culturally responsive is by leveraging the natural way that students learn. Our students do not come to school as a blank slate, they have life experiences and funds of knowledge that can be an asset in the mathematics classroom.

For example, sometimes we are in a rush to teach students shortcuts such as the standard algorithm for addition, and students struggle because it doesn’t make sense. If you ask a first grade what is 12 and 5. They might say 5, 10, 15, 16, 17. They might come from a culture where music is important and they might be used to songs, rhymes, and chants that make skip counting an intuitive strategy. However, as soon as students are taught that the standard algorithm is the right way and don’t take into account the strategies they already knew, students will line up digits and only use a method that is not always the most effective. How can we find out what strategies and methods students already use? What are some common effective strategies that are employed by groups of students in our classroom?

In order to be culturally responsive, facilitated lessons must be composed of challenging and complex work. Our students need to be constantly engaging in rigorous work, and it cannot be watered down. Rigorous work does not mean more work. It does not “drill and kill” by having students complete numerous pages of simple word problems, multiplication facts, and computation worksheets. Students need to take part in tasks and lessons that engage them in conceptual and procedural understandings. For example, 3 Act Tasks prompt students to analyze and solve a real-life problem by using their estimation skills and collaborating with their peers. Routines such as Which One Does Not Belong prompt students to make connections to real life, their culture, math concepts, and other content areas.

Going beyond having a growth mindset is necessary to having a culturally responsive mathematics classroom. Students have grit and are capable. What might be missing is the structure that allows students to feel challenged, safe to share their thinking, and value their voice. Independent learning stations or workstations allow students time to work on their own or with a partner without the teacher. Students take ownership of their own learning by applying and practicing what they have learned. Flipgrid is a free online tool that allows students to record their thinking to a prompt or task, and then students can respond to each other. This type of activity helps students feel valued, and they gain confidence hearing feedback from their peers.

Thanks to Chiquita, Autumn, James, and Cindy for their contributions!

(This is the first post in a two-part series)

The new question-of-the-week is:

What are specific ways educators can make teaching math more culturally responsive?

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5 steps to a problem-solving classroom culture

Math problems can be engaging and thought-provoking with the right instructional strategies

Creating a culture of problem solving in a math classroom or in a school involves prompting students and educators to think a little differently and systemically.

“The world does not need more people who are good at math,” said Gerald Aungst , supervisor of gifted and elementary mathematics in Pennsylvania’s Cheltenhamn Township Schools. “What the world needs are more problem solvers and more innovators.”

“We want people who are innovators, and don’t assume that what people tell them is impossible is impossible,” Aungst said during an edWeb leadership webinar .

One of the most important mindsets comes in realizing that, even in math, the context of a statement makes all the difference. Students should understand more than just the mechanics of math, Aungst said—they should investigate the context, the meaning, and how math problems and concepts work in a particular situation.

The five steps to building a problem-solving culture aren’t quick fixes or easy tips, Aungst said, but can be impactful when applied with the bigger picture of the classroom environment in mind.

( Next page: Five steps to creating a culture of problem solving )

“Our need to have things explained is as strong an impulse in our kids, and in us, as being hungry and thirsty,” Aungst said. “The problem with how we usually teach math is that we take all that wondering away.”

Educators usually teach math by laying out the facts, showing them processes, and asking students to practice until they achieve “mechanical perfection”–students have nothing to wonder about.

“One element of conjecture is being able to provoke that sense of wonder in kids, and allowing them to look for explanations and let that drive keep them engaged,” Aungst said.

But it goes deeper than that, he said.

“It’s about students not just solving problems–it’s about them looking for problems, too,” he added. “Innovators are looking for problems and they try to solve them before anyone even realizes the problem exists. We need innovators. Math class is a great place to start doing that.”

Educators should strive to avoid ending with the answer. Instead, they should ask students why they think the answer is what it is, how they arrived at the answer, if other answers are possible, if other methods of solving are possible, if students encountered difficulty, and if so, how they overcame it.

Digital tools to support conjecture include: http://data.gov http://edte.ch/blog/maths-maps http://www.geogebra.org

Communication

When students are able to explain their thought processes and understanding, their own knowledge increases.

One way to promote better math learning is to think of math as if it were a foreign language.

“If all we’re doing is teaching students how to move the symbols around and get an answer out of it, without embedding meaning into that, then the meaning behind the math is completely lost,” he said. “Learning how to do math is like learning how to read a foreign language.”

Students should be able to explain, in their own words, what numbers and symbols mean and represent.

Instead of asking students to show their work, ask them to convince mathematical experts that their solution is a good one–students understand what they do, but communicating it to someone else is a challenge.

Digital tools for communication include: Infographics such as http://piktochart.com and http://infogr.am Social media (speaking to others about the math students are doing) YouTube and Vine Classroom blogs

Collaboration

“Problem solving in the real world is nearly always collaborative,” Aungst said. “In fact, competition might even serve to dampen innovation. We want to get our kids working together.”

Working together inspires students to consider other points of view and other approaches to problems. This, in turn, informs, and may change, their thinking.

Educators could begin with a “You, Y’all, We” approach: present the problem first, and let students work on that problem individually. They’ll struggle, Aungst said, but that’s OK. Then, move to small-group discussion, before involving the whole class in the discussion or in solving the problem.

Aungst also recommends the “three before me” strategy, in which students consult three other resources or people before bringing an “unsolvable” problem to their teacher.

Digital tools for collaboration and building classroom teams include: Wikis and Google Sites Google Classroom Skype and Google Hangouts Wiggio Edmodo

As odd as it seems, chaos promotes learning and discovery, Aungst said.

“What it really is about is the fact that problem solving is messy–it’s not a linear step-by-step,” he said. “Real world problem solving is a messy thing.”

Students should struggle in productive ways, and if they’re not, instruction isn’t particularly effective. In short, they need “cognitive sweat,” Aungst said.

Digital tools to support chaos include: http://enlvm.usa.edu http://ohiorc.org/for/math/stella http://mathpickle.com

Celebration

Educators should celebrate students’ growth, successes, “and even their failures, and what you can learn from their failures,” Aungst said.

Sometimes, a “catch me if you can” strategy works well. Educators tell students they plan to make mistakes, and students must try to identify those mistakes. This makes it safe for students to point out errors.

“It’s really important that you validate effort, and not answers,” he said. “It’s really important that we recognize that the students who start out as the smartest at the beginning of the year may not be the smartest at the end of the year.”

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Promoting Inclusion in Mathematics Assessment: Applying a culturally responsive lens in mathematics assessment

More mirrors, please.

When a story on the  radio about a new education startup designed for Black students featured the opening synthesizer sounds from Beyonce’s “Formation," I was immediately hooked. 1 So much so that I immediately had to google the full version of the video, Lit in Formation , that includes lyrics about the “lack of diverse texts” in education and how “I like my books to look like me, mirrors and windows,” 2 which references the work of Rudine Sims Bishop. 3  

When I reflect on what my excitement was telling me, I think about the first time in my educational experience that I had a teacher who “looked like me”—she identifies as being from South Asia and I identify as Southeast Asian. It was not until I was   an undergraduate in college.   Similarly, I think about the first time   I learned about   non-European mathematicians, which was in a History of Mathematics course while pursuing my bachelor’s degree in mathematics .  

And so, those lyrics were   another signal pointing to a growing appetite for education materials that reflected a wider range of people, experiences, and identities, including communities of people who look like me.  

What's the big deal about culturally responsive education? 

The question of how to disrupt inequities in U.S. public education, often referred to as “achievement gaps” and “opportunity gaps,” is a common topic of discussion among educators. The disparities in the educational experiences of  Black, Brown , and  Indigenous students have been documented going back to the 1970s. 4,5  

When I use the term Brown, I am referring to people who identify as Latinx, Asian, and/or Pacific Islander. Students who identify as Asian and Pacific Islander are often erased from educational inequity narratives because of a myth that there is an Asian Advantage , especially in mathematics. 6 That myth is harmful. It masks the tremendous diversity among Asian Americans and Pacific Islanders, who may have cultural ties to at least one of 62 Asian or Pacific Island nations, and ignores the   numerous disparities experienced by members of the community. 7,8  

But why do these disparities persist?  

In mathematics, one reason is that there is a philosophy that teaching and learning mathematics is culturally neutral. Some mathematics educators may even argue that the beauty of mathematics is in its objective, abstract, and apolitical nature. However, this perspective only serves to maintain the educational inequities experienced by Indigenous, Brown, and Black students in our country.  

Fortunately, paradigms have shifted in recent decades. There is greater awareness and acceptance that mathematics education is a cultural process, and students bring with them knowledge and understanding filtered through their individual cultural lens. Based on the pedagogical  framework by Dr. Gloria Ladson-Billings, “culturally relevant teachers utilize students’ culture as a vehicle for learning. 9 Furthermore, culturally relevant teaching “ empowers students—intellectually, socially, emotionally, and politically—by using cultural referents to impart knowledge, skills, and attitudes.” 10 As a side note, I use the term culturally responsive, which   shares similar goals with culturally relevant, but places a larger   emphasis on strategies and practices . 11  

While a framework for approaching education with a culturally responsive lens is not new, there is an untapped component of mathematics education that deserves greater focus and attention—that is, a focus on designing mathematics curriculum and assessments to be more inclusive, representative, and responsive to an   increasingly diverse   U.S. public school student population. 12  

Benefits for students

One of the most exciting aspects of applying a culturally responsive lens in mathematics is that it  rehumanizes mathematics   by “conjuring up feelings of joy.” 13 It could be joy that is felt when making a connection to a familiar celebration or type of food that is referenced in a mathematics problem. Or the subtle joy of seeing a student’s name in a mathematics task that is similar to names commonly used in your community. Or even a sense of belonging that is invoked when seeing an image of a student that reflects your own identity.  

However large or small that joy may be, implementing principles of culturally responsive education allows students the opportunity to make personal connections to mathematics content, which is linked to the following positive student outcomes :  

improved student interest and enjoyment in math,  

increased engagement,  

persistence in problem-solving,  

confidence in mathematics,  

and increased mathematics performance on formal assessments. 14

Studies have also shown that culturally responsive approaches in mathematics encourage students from all backgrounds to use their higher-level thinking skills, such as analysis, reasoning, and evaluation (e.g., CCSS math practices). 15  

Returning to the analogy of  mirrors and windows   in education, when we create curriculum and assessments that reflect the cultural identities of students of color, these “mirrors” serve to validate and celebrate students’ racial backgrounds and experiences. At the same time, curriculum and assessments can also serve as “windows” for students of different cultural backgrounds, providing opportunities for each student to understand his, her, or their connection to all other humans. 3  

In short, designing culturally responsive mathematics assessments benefits all students.  

CenterPoint's approach for applying culturally responsive practices in assessment

Our commitment  .

As assessment developers, we know that students of color make up  53% of the student population in U.S. public schools, and that percentage has been increasing since 2017 . 16,17  

Knowing this, CenterPoint is committed to developing culturally responsive mathematics assessments that include a diverse representation of students’ cultural backgrounds, experiences, and communities, which may differ from our own lived experiences and identities. By shifting away from a culturally neutral mathematics paradigm and building upon the cultural capital of students, we promote inclusion and add meaning to mathematics for students.  

What about social justice? 

A prominent feature of culturally responsive education is that the problems and tasks connect to real-world problems for which students are asked to consider solutions, thereby promoting the development of students’ critical consciousness. These issues may involve injustices that exist in students’ communities or nationwide. Specifically, culturally responsive mathematics tasks   are real-world problems that take into account a broader world than the typically Eurocentric one emphasized in the past. They may ask students to “make sense   of the world through mathematics” and “critique society—that is, make empowered decisions about themselves, communities, and world .” 18  

Despite the importance of empowering students to challenge injustice, including social justice topics (e.g., voting rights, climate justice, income gaps, food insecurity, racial injustice) in assessment can be problematic from a bias perspective and therefore raise fairness concerns. So, unless it is essential for valid measurement, we avoid references to topics that may cause students to have strong, emotional reactions that distract or prevent them from completing a problem or task .  Instead, these issues are more appropriate as topics of discussion in mathematics classrooms.  

CenterPoint’s checklist 

When developing culturally responsive assessments in mathematics, we strive to write items with engaging contexts that center the diverse cultures of students of color, which have been historically underrepresented in U.S. education.   

Specifically, we consider the following criteria when developing a culturally responsive assessment item (i.e., test question or task) in mathematics :  

The cultural content of the item is interesting and promotes connections.

The cultural content of the item is accurate.  

The cultural content shows appreciation of the culture and avoids the use of stereotypes.  

Students can answer the question(s) with the information provided.  

The item has been written in simple, plain language.    

Sample items

Let’s consider two sample items, one that meets and one that does not meet CenterPoint’s criteria for culturally responsive math assessment items. Both items are designed to measure the Common Core State Standard 3.OA.A.3 (i.e., use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities).  

Figure 1 . Two items aligned to content standard 3.OA.A.3  

Item A is an example of an item that meets CenterPoint’s criteria for culturally responsive math items.  

The cultural content of the item is interesting and promotes connections.  Featuring lemongrass, which is typically used as an herb in South Asian and Southeast Asian cooking and medicine, centers the content of the item around a diverse set of identities, thereby decentering  White, European cultures . Students familiar with South Asian and Southeast Asian cooking or medicinal practices are likely to make a positive connection to the item based on personal experiences tasting recipes or using remedies with lemongrass. Students who are not familiar with the herb may also be intrigued by the unique ingredient.  

Cultural content of the item is accurate.  Lemongrass is a  perennial grass  native to Sri Lanka and South India and is  used as an herb in cooking and medicine. 19,20  

The cultural content shows appreciation of the culture and avoids the use of stereotypes.  Lemongrass is cultivated throughout Asia and other tropical regions. It is a common ingredient in many Indian, Vietnamese, Thai, and Indonesian recipes and can also be used in a variety of healing therapies. The item avoids any negative portrayals of the character, Raj.  

Students can answer the question(s) with the information provided.  A sufficient amount of context is provided, in particular with the inclusion of art, for students unfamiliar with lemongrass. Students who are not familiar with lemongrass will not be disadvantaged by the reference to a new kind of plant.  

The item has been written in simple, plain language.  Though lemongrass is often referred to as having stalks, the word “stems” was chosen to ensure the vocabulary is appropriate for 3 rd grade students. Other key phrases in the item, such as “tie up” and “bunch,” are also grade-level appropriate. 21 [Note: Although it is a common practice in textbooks and print media to italicize foreign words, we intentionally do not follow this convention in our mathematics assessments. We feel this practice further prioritizes mainstream U.S. culture and reinforces the “otherness” of non-English languages and the communities who speak them.]  

In contrast, Item B was not designed to meet our culturally responsive item criteria. The cultural context, pencils being placed into bags or other containers, is frequently used in mathematics curricula but not intentionally culturally specific. Thus, Item B fails to amplify a typically underrepresented culture, student experience, or cultural identity.  

Item B is easily accessible to all, valid, and appropriate for use with students. It is the kind of item students encounter all the time. It just wasn't designed to be culturally responsive.  

Closing thoughts

Do we think that one question about lemongrass on a mathematics test is going to dramatically change students’ lives and develop their critical consciousness? Of course not. But having grown up in a predominantly White community, I would have appreciated having a wider variety of cultural backgrounds, experiences, and communities represented in the curriculum. So if we can plant some seeds that affirm the cultures and identities of Brown, Indigenous, and Black students, then we can begin to honor each other’s humanity using mathematics.   

Not only that but creating assessments that are inclusive is part of  who we are .  

We would love to hear about your experience with culturally responsive assessments and/or curricula!  

Do you implement culturally responsive strategies in your teaching or assessment practices?   

Do CenterPoint’s criteria for designing culturally responsive assessment items in mathematics resonate with you? What do you appreciate the most or what might you do differently?  

If you’re up for sharing, when was the first time you recognized yourself reflected in the curriculum? How did that make you feel?

Acknowledgements

CenterPoint would like to thank Nirupa Mathew and the mathematics editorial team at  Curriculum Associates  for their partnership and guidance in navigating culturally responsive assessment design in mathematics.  

Katrina Santner serves as a Senior Instructional Designer in   Mathematics   at   CenterPoint Education Solutions .  CenterPoint is a mission-driven, nonprofit organization dedicated to supporting schools and districts in implementing coherent instructional models consisting of high-quality curriculum, tightly aligned assessments, and professional learning.   

[1] Kamenetz, A. (2021, February 25).  Teaching students a new Black history . NPR.  https://www.npr.org/2021/02/25/964418856/teaching-students-a-new-black-history  

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Innovative and Powerful Pedagogical Practices in Mathematics Education

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• Jodie Hunter 7 ,
• Jodie Miller 8 ,
• Ban Heng Choy 9 &
• Roberta Hunter 7  

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Powerful and innovative pedagogical practices are necessary for all students to learn mathematics successfully and equip them for the future. In this chapter, we review Australasian studies that provide evidence of pedagogical practices that support creative and flexible mathematical thinkers for the 21st century. The review is structured around three key themes that were evident in the research literature. The first theme is the need to develop innovative learning environments that benefit all learners. The second theme is centred on how both tasks and tools can be used to support powerful pedagogical practices. Finally, the third theme reviews the challenges of developing innovative mathematical learning environments. We argue for the need for effective pedagogy for all learners and a need for ambitious, future-focused teaching in mathematics education.

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Hunter, J., Miller, J., Choy, B.H., Hunter, R. (2020). Innovative and Powerful Pedagogical Practices in Mathematics Education. In: Way, J., Attard, C., Anderson, J., Bobis, J., McMaster, H., Cartwright, K. (eds) Research in Mathematics Education in Australasia 2016–2019. Springer, Singapore. https://doi.org/10.1007/978-981-15-4269-5_12

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DOI : https://doi.org/10.1007/978-981-15-4269-5_12

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Volume III, Issue IV

Keys to Productive Discussions in the Math Classroom

To listen well is as powerful a means of influence as to talk well and is as essential to all true conversation.

– Chinese Proverb

A challenge faced by math educators of all levels is how to engage students in their mathematical content through rich discussion or discourse. In classrooms where there is high-quality mathematical discourse, teachers and students ask challenging and thought-provoking questions, and there is skillful facilitation of meaningful discussions focused on the mathematics. The discussions emphasize reasoning, proof, evaluation, and justification. Students learn from one another and value the thinking of their peers. The focus of the conversation is not simply the answer to the problem, but also the students’ strategies, discoveries, conjectures, and reasoning.

The third Standard for Mathematical Practice places a strong emphasis on meaningful discourse. In this standard, students are expected to construct viable arguments and critique the reasoning of others. Meaningful discussions in the mathematics classroom rely on purposeful instructional moves from the teacher, as well as a clear understanding of the demands that are placed on students. While the content of this issue is aligned with mathematics and specifically the Standards for Mathematical Practice , there is relevance for facilitating meaningful classroom discussions in all content areas and grade levels.

The Common Core places a strong emphasis on mathematical reasoning and deep content understanding. Creating the right conditions for these discussions and facilitating conversations that emphasize a deep study of the mathematics is a challenging task. The following keys can help teachers ensure that the discourse in their mathematics classrooms is rich and extends the learning of students. The single most important thing teachers should do to ensure the success of discussions is to ask meaningful questions and facilitate the dialogue among students. The goal in any mathematical discussion is to support the students’ in constructing viable arguments and critiquing the reasoning of others.

Keys for Preparing for the Discussion

Anticipate the strategies students might use, how they will represent their thinking, and predict students’ misconceptions. In addition to drawing on their knowledge of mathematical content, teachers must also bring to classroom discussions an understanding of their students’ prior knowledge and experiences. Once the task has been designed, the teacher must be ready to handle the different strategies that the students will propose. One way to prepare is to draft all possible student strategies, prioritize how those will be shared with the class, and anticipate places where there may be flaws in students’ thinking or misconceptions. By making these predictions in advance of the class discussion, teachers will have a clear sense of the critical “look-fors” as the students are working and an idea of how they wish to shape the classroom discussion. Undoubtedly, students will come up with strategies that the teacher has not predicted; however, teachers will be far more prepared to make sense of these approaches to problem solving when they have thought ahead about what students might bring to the experience.

For instance, consider the following problem:

Anna is collecting pennies for a school-wide penny drive. She has 357 pennies saved in the first week and 225 pennies saved in the second week. Her goal is to donate 1,000 pennies. How many more pennies will Anna need to reach her goal?

Plan questions that will guide students in answering both how they solved a problem and why they chose the solution they used. Preplanning thought-provoking questions will ensure a high level of intellectual engagement during the lesson. Including the context of the problems is essential when forming these questions. By asking students to use the context of the problem when determining their solutions, they are more likely to have solid reasoning for why they solved the problem in the way that they did. For instance, the teacher might ask:

• Why did you _____________ when the problem asked for _____________?
• What does _________ mean in terms of _________________ as it stated in the problem?
• Does this solution make sense given what the problem is asking?
• Why are we ______________ in this problem?

Decide which strategies should be prioritized when sharing with the whole class. It can be overwhelming for students to hear and understand the reasoning behind too many different strategies at once. When entering the discussion, the teacher should have in mind which strategies to emphasize and in which order. For instance, if it is a problem dealing with subtraction, the teacher may choose to emphasize the use of an unmarked number line or adding up before having discussions about adding or subtracting the same number from the minuend and subtrahend in order to create an easier problem and not change the answer.

Keys for Facilitating Discussion

Establish a safe environment where students can take risks and where there are norms for classroom discussions. In order for students to openly share their thinking and risk-making mistakes in front of their peers, it is imperative that there is a supportive classroom environment. Everyone should understand their role in the classroom through the development of classroom norms. The teacher is expected to pose thought-provoking questions, support students’ conversations, listen carefully to monitor students’ understanding and misconceptions, encourage student participation in discussions, and promote student reflection about the learning experience.

Nancy Anderson, one of the authors of the National Council of Teacher of Mathematics’ book entitled, Classroom Discussions: Using Math Talk to Help Students Learn , suggests that teachers instruct their students on the importance of and expectations for mathematical conversations at the start of the school year. She explains how talking like mathematicians can enable students to be stronger mathematical thinkers. As Anderson tells her students:

• Talking and thinking together can help all students understand math better
• It is necessary for more than one person to help solve challenging problems
• There is a great deal to be learned from listening to how other’s think
• Talking about your thinking helps you to clarify your own thoughts
• When talking about the mathematics, you practice using important math vocabulary
• You can learn a great deal about what it takes to understand the ideas of others.

Along with establishing a rationale for mathematical discussions, it is also critical to establish expectations for respectful listening. Students need to be seated where they can see and hear the speaker, and they are expected to listen actively and be prepared to respond to the ideas of others. Students are taught how to respectfully disagree and question one another. Above all, there is acceptance of all ideas and all contributions to the discussion are honored. Once the school year is under way, it is important to revisit the established norms in order to maintain the quality of conversations.

Teach students the expectations for classroom discussions. Despite efforts to establish a rationale for discussions and expectations for listening, rich discussions in mathematics do not happen by chance. The explicit teaching of how students are expected to respond and interact during a classroom discussion in mathematics is necessary. students sharing their thinking should know that their explanations require more than just a description of the strategy they used to solve a problem. Rather, students need to include some sort of visual representation, along with an explanation of how they solved the problem and why they chose to solve the problem in that way.

Students who are listening should be attentive to the thinking of others, reflect on the ideas they have heard to evaluate their efficiency, determine if they agree or disagree, if they understand the thinking of their peers, and what similarities and differences they see between their own thinking and the thinking of others. Students need to be taught how to agree and disagree and how to ask questions for clarification. Provide students with prompts to use during discussions. For instance, students might say:

• The way ______ solved the problem makes sense to me because…
• ______’s strategy was similar to mine because…
• ______’s strategy was different than mine because…
• What I don’t understand about ______’s explanation is why _______.
• I will need to hear _______ explain how _________ again.
• Why did you _____ when you were solving this problem?
• I understand how you ______, but why did you ______?

Present meaningful problems. Teachers should focus on assigning mathematical tasks that are appropriately challenging and enhance students’ learning. Mathematical tasks should investigate important mathematical ideas and have authentic contexts and relevance for students. The problems posed should have multiple solution strategies, encourage investigation, promote reasoning, and require students to provide justifications for their thinking. Ultimately, mathematical tasks should be worthy of student discussion and emphasize important mathematical concepts.

Build in opportunities for independent work and partner or small group work. In order to help students summarize and understand their thinking as well as the thinking of others, it is essential to provide opportunities for students to “turn and talk” about their ideas. For instance, after presenting a problem, students may be asked to represent or state in their own words what the problem is asking, then share that with a partner. After finding an entry point and solving a problem independently, students should share their strategies with a partner or in a group, prior to sharing with the whole class. This gives students practice constructing arguments, providing justifications, and critiquing the thinking of others. Students learn how to listen in a way that prepares them to restate their partner’s thinking in their own words, as well as listening to understand and pose questions of their partner. Partnerships ensure a higher level of accountability and student engagement than is possible with only whole class discussions.

Facilitate the sharing of strategies with the whole class. While students are engaged in discussion, it is the teacher’s role to promote students’ reasoning, ensure that multiple solutions and answers are considered, hold students accountable for sharing both how they solved a problem and why they solved it using a specific strategy, and to make sure that students are actively listening and responding to each other. Teachers can do this is through the use of meaningful questions that will support and extend students’ understanding of the reasoning of others, along with the important mathematical ideas.

The teacher should begin by collecting all students’ answers and encouraging students to think about whether or not more than one answer could be correct given the context of the problem. Then, as chosen students defend their solutions and share arguments for their strategies, the teacher ensures active listening and reflection through the use of guiding questions. For instance, the teacher might ask the student who is sharing:

• Why did you ______?
• Where did ______ come from?
• Where are the original numbers in the problem?
• How did you represent your thinking?

Questions to ask the rest of the class might be:

• Could somebody repeat what _____ has shared in their own words?
• Is _______’s reasoning reasonable? Why or why not?
• Is _______’s strategy an efficient way to solve this problem? Why or why not?
• Do you agree with ______? Why or why not?
• What do you wonder after hearing ______’s thinking?
• Can you think of a counter example? In other words, can you think of an example that would disprove an idea that has been presented?

Promote student reflection on the different strategies. A powerful instructional move after students have heard the thinking of others is to send them back to work in partners or in small groups to reflect on the arguments of others. Carefully crafted questions such as the following can help guide these discussions:

• Which strategy have you heard is the most efficient for solving this problem? Why?
• What are some similarities you have seen between the strategies being used? What are some differences?
• What new ideas did you hear today?
• What confused you?
• What strategies do you think you could try when solving future problems?

In summary, how successfully a teacher facilitates a discussion drives how mathematically rigorous the work is for students. In order for students to be successful with constructing viable arguments and critiquing the reasonableness of answers, students need ample practice solving problems in a variety of ways and defending their thinking with others. Equally important is that students know how to listen to the thinking of others, and pose questions and counter examples as a way of deepening their mathematical understanding. The success of these small and large group discussions rests on the ability of the teacher to plan thoughtfully and facilitate purposefully.

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Please include the following citation on all copies:

Clayton, Heather. “Keys to Productive Discussions in the Math Classroom.” Making the Common Core Come Alive! Volume III, Issue IV, 2014. Available at www.justaskpublications.com. Reproduced with permission of Just ASK Publications & Professional Development (Just ASK). ©2014 by Just ASK. All rights reserved.

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Mathematics as a Complex Problem-Solving Activity

By jacob klerlein and sheena hervey, generation ready.

By the time young children enter school they are already well along the pathway to becoming problem solvers. From birth, children are learning how to learn: they respond to their environment and the reactions of others. This making sense of experience is an ongoing, recursive process. We have known for a long time that reading is a complex problem-solving activity. More recently, teachers have come to understand that becoming mathematically literate is also a complex problem-solving activity that increases in power and flexibility when practiced more often. A problem in mathematics is any situation that must be resolved using mathematical tools but for which there is no immediately obvious strategy. If the way forward is obvious, it’s not a problem—it is a straightforward application.

Mathematicians have always understood that problem-solving is central to their discipline because without a problem there is no mathematics. Problem-solving has played a central role in the thinking of educational theorists ever since the publication of Pólya’s book “How to Solve It,” in 1945. The National Council of Teachers of Mathematics (NCTM) has been consistently advocating for problem-solving for nearly 40 years, while international trends in mathematics teaching have shown an increased focus on problem-solving and mathematical modeling beginning in the early 1990s. As educators internationally became increasingly aware that providing problem-solving experiences is critical if students are to be able to use and apply mathematical knowledge in meaningful ways (Wu and Zhang 2006) little changed at the school level in the United States.

“Problem-solving is not only a goal of learning mathematics, but also a major means of doing so.”

(NCTM, 2000, p. 52)

In 2011 the Common Core State Standards incorporated the NCTM Process Standards of problem-solving, reasoning and proof, communication, representation, and connections into the Standards for Mathematical Practice. For many teachers of mathematics this was the first time they had been expected to incorporate student collaboration and discourse with problem-solving. This practice requires teaching in profoundly different ways as schools moved from a teacher-directed to a more dialogic approach to teaching and learning. The challenge for teachers is to teach students not only to solve problems but also to learn about mathematics through problem-solving. While many students may develop procedural fluency, they often lack the deep conceptual understanding necessary to solve new problems or make connections between mathematical ideas.

“A problem-solving curriculum, however, requires a different role from the teacher. Rather than directing a lesson, the teacher needs to provide time for students to grapple with problems, search for strategies and solutions on their own, and learn to evaluate their own results. Although the teacher needs to be very much present, the primary focus in the class needs to be on the students’ thinking processes.”

(Burns, 2000, p. 29)

Learning to problem solve

To understand how students become problem solvers we need to look at the theories that underpin learning in mathematics. These include recognition of the developmental aspects of learning and the essential fact that students actively engage in learning mathematics through “doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning” (Copley, 2000, p. 29). The concept of co-construction of learning is the basis for the theory. Moreover, we know that each student is on their unique path of development.

Beliefs underpinning effective teaching of mathematics

• Every student’s identity, language, and culture need to be respected and valued.
• Every student has the right to access effective mathematics education.
• Every student can become a successful learner of mathematics.

Children arrive at school with intuitive mathematical understandings. A teacher needs to connect with and build on those understandings through experiences that allow students to explore mathematics and to communicate their ideas in a meaningful dialogue with the teacher and their peers.

Learning takes place within social settings (Vygotsky, 1978). Students construct understandings through engagement with problems and interaction with others in these activities. Through these social interactions, students feel that they can take risks, try new strategies, and give and receive feedback. They learn cooperatively as they share a range of points of view or discuss ways of solving a problem. It is through talking about problems and discussing their ideas that children construct knowledge and acquire the language to make sense of experiences.

Students acquire their understanding of mathematics and develop problem-solving skills as a result of solving problems, rather than being taught something directly (Hiebert1997). The teacher’s role is to construct problems and present situations that provide a forum in which problem-solving can occur.

Why is problem-solving important?

Our students live in an information and technology-based society where they need to be able to think critically about complex issues, and “analyze and think logically about new situations, devise unspecified solution procedures, and communicate their solution clearly and convincingly to others” (Baroody, 1998). Mathematics education is important not only because of the “gatekeeping role that mathematics plays in students’ access to educational and economic opportunities,” but also because the problem-solving processes and the acquisition of problem-solving strategies equips students for life beyond school (Cobb, & Hodge, 2002).

The importance of problem-solving in learning mathematics comes from the belief that mathematics is primarily about reasoning, not memorization. Problem-solving allows students to develop understanding and explain the processes used to arrive at solutions, rather than remembering and applying a set of procedures. It is through problem-solving that students develop a deeper understanding of mathematical concepts, become more engaged, and appreciate the relevance and usefulness of mathematics (Wu and Zhang 2006). Problem-solving in mathematics supports the development of:

• The ability to think creatively, critically, and logically
• The ability to structure and organize
• The ability to process information
• Enjoyment of an intellectual challenge
• The skills to solve problems that help them to investigate and understand the world

Problem-solving should underlie all aspects of mathematics teaching in order to give students the experience of the power of mathematics in the world around them. This method allows students to see problem-solving as a vehicle to construct, evaluate, and refine their theories about mathematics and the theories of others.

Problems that are “Problematic”

The teacher’s expectations of the students are essential. Students only learn to handle complex problems by being exposed to them. Students need to have opportunities to work on complex tasks rather than a series of simple tasks devolved from a complex task. This is important for stimulating the students’ mathematical reasoning and building durable mathematical knowledge (Anthony and Walshaw, 2007). The challenge for teachers is ensuring the problems they set are designed to support mathematics learning and are appropriate and challenging for all students.  The problems need to be difficult enough to provide a challenge but not so difficult that students can’t succeed. Teachers who get this right create resilient problem solvers who know that with perseverance they can succeed. Problems need to be within the students’ “Zone of Proximal Development” (Vygotsky 1968). These types of complex problems will provide opportunities for discussion and learning.

Students will have opportunities to explain their ideas, respond to the ideas of others, and challenge their thinking. Those students who think math is all about the “correct” answer will need support and encouragement to take risks. Tolerance of difficulty is essential in a problem-solving disposition because being “stuck” is an inevitable stage in resolving just about any problem. Getting unstuck typically takes time and involves trying a variety of approaches. Students need to learn this experientially. Effective problems:

• Are accessible and extendable
• Allow individuals to make decisions
• Promote discussion and communication
• Encourage originality and invention
• Encourage “what if?” and “what if not?” questions
• Contain an element of surprise (Adapted from Ahmed, 1987)

“Students learn to problem solve in mathematics primarily through ‘doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning.”

(Copley, 2000, p. 29)

“…as learners investigate together. It becomes a mini- society – a community of learners engaged in mathematical activity, discourse and reflection. Learners must be given the opportunity to act as mathematicians by allowing, supporting and challenging their ‘mathematizing’ of particular situations. The community provides an environment in which individual mathematical ideas can be expressed and tested against others’ ideas.…This enables learners to become clearer and more confident about what they know and understand.”

(Fosnot, 2005, p. 10)

Research shows that ‘classrooms where the orientation consistently defines task outcomes in terms of the answers rather than the thinking processes entailed in reaching the answers negatively affects the thinking processes and mathematical identities of learners’ (Anthony and Walshaw, 2007, page 122).

Effective teachers model good problem-solving habits for their students. Their questions are designed to help children use a variety of strategies and materials to solve problems. Students often want to begin without a plan in mind. Through appropriate questions, the teacher gives students some structure for beginning the problem without telling them exactly what to do. In 1945 Pólya published the following four principles of problem-solving to support teachers with helping their students.

• Understand and explore the problem
• Find a strategy
• Use the strategy to solve the problem
• Look back and reflect on the solution

Problem-solving is not linear but rather a complex, interactive process. Students move backward and forward between and across Pólya’s phases. The Common Core State Standards describe the process as follows:

“Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary”. (New York State Next Generation Mathematics Learning Standards 2017).

Pólya’s Principals of Problem-Solving

Students move forward and backward as they move through the problem-solving process.

The goal is for students to have a range of strategies they use to solve problems and understand that there may be more than one solution. It is important to realize that the process is just as important, if not more important, than arriving at a solution, for it is in the solution process that students uncover the mathematics. Arriving at an answer isn’t the end of the process. Reflecting on the strategies used to solve the problem provides additional learning experiences. Studying the approach used for one problem helps students become more comfortable with using that strategy in a variety of other situations.

When making sense of ideas, students need opportunities to work both independently and collaboratively. There will be times when students need to be able to work independently and other times when they will need to be able to work in small groups so that they can share ideas and learn with and from others.

Getting real

Effective teachers of mathematics create purposeful learning experiences for students through solving problems in relevant and meaningful contexts. While word problems are a way of putting mathematics into contexts, it doesn’t automatically make them real. The challenge for teachers is to provide students with problems that draw on their experience of reality, rather than asking them to suspend it. Realistic does not mean that problems necessarily involve real contexts, but rather they make students think in “real” ways.

Planning for talk

By planning for and promoting discourse, teachers can actively engage students in mathematical thinking. In discourse-rich mathematics classes, students explain and discuss the strategies and processes they use in solving mathematical problems, thereby connecting their everyday language with the specialized vocabulary of mathematics.

Students need to understand how to communicate mathematically, give sound mathematical explanations, and justify their solutions. Effective teachers encourage their students to communicate their ideas orally, in writing, and by using a variety of representations. Through listening to students, teachers can better understand what their students know and misconceptions they may have. It is the misconceptions that provide a window into the students’ learning process. Effective teachers view thinking as “the process of understanding,” they can use their students’ thinking as a resource for further learning. Such teachers are responsive both to their students and to the discipline of mathematics.

“Mathematics today requires not only computational skills but also the ability
to think and reason mathematically in order to solve the new problems and learn the new ideas that students will face in the future. Learning is enhanced in classrooms where students are required to evaluate their own ideas and those of others, are encouraged to make mathematical conjectures and test them, and are helped to develop their reasoning skills.”

(John Van De Walle)

“Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students.”

How teachers organize classroom instruction is very much dependent on what they know and believe about mathematics and on what they understand about mathematics teaching and learning. Teachers need to recognize that problem-solving processes develop over time and are significantly improved by effective teaching practices. The teacher’s role begins with selecting rich problem-solving tasks that focus on the mathematics the teacher wants their students to explore. A problem-solving approach is not only a way for developing students’ thinking, but it also provides a context for learning mathematical concepts. Problem-solving allows students to transfer what they have already learned to unfamiliar situations. A problem-solving approach provides a way for students to actively construct their ideas about mathematics and to take responsibility for their learning. The challenge for mathematics teachers is to develop the students’ mathematical thinking process alongside the knowledge and to create opportunities to present even routine mathematics tasks in problem-solving contexts.

Given the efforts to date to include problem-solving as an integral component of the mathematics curriculum and the limited implementation in classrooms, it will take more than rhetoric to achieve this goal. While providing valuable professional learning, resources, and more time are essential steps, it is possible that problem-solving in mathematics will only become valued when high-stakes assessment reflects the importance of students’ solving of complex problems.

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In this article, Dr Sue Gifford outlines how we can create positive attitudes and higher achievement in mathematics, starting in the Early Years.

New Building Blocks: a Review of the Pilot Early Learning Goals

This article discusses the revised Early Learning Goals for mathematics which were announced in June 2018.

Young Children's Mathematical Recording

In this article, Janine Davenall reflects on children's personalised mathematical recordings as part of a small research project based in her Reception class.

Maths Trails - Encouraging Purposeful Outdoor Learning

In this article, Becky Moseley outlines key considerations for Primary teachers wanting to create a maths trail in their own locality.