Let’s Be Clear: One Teacher’s Sensemaking About How to Make Sense

Authors note: This article was co-written by Meghan Odsliv Bratkovich, and was submitted for publication to the National Council of Teachers of Mathematics research journal.

How students have been taught mathematics has changed over time. Algorithmic computation and formulaic memorization have largely given way to divergent thinking and emphasis on mathematical understanding.  The Common Core State Standards both reflect and guide corresponding shifts in the field of mathematics and in mathematics teaching, emphasizing communication, argumentation, and reasoning skills that are essential for college and career readiness.  Despite this, even highly successful teachers often lack the time and space necessary to make sense of what this means and ways it might positively affect their practice–how students are expected to express, construct, and represent mathematical sense, how teachers are expected to make sense of mathematics for their students, and how teachers’ engagement in systematic sensemaking through inquiry can facilitate this process.

The interconnected considerations incorporated in Practice Standard 3 are integral to the construction, communication, and critiques of viable arguments in mathematics (e.g., appropriate processes, structures, proofs, formats, logic, and terms). Even very knowledgeable and highly effective teachers can leave these underlying strands of argumentation snarled between and knotted up in correct and incorrect solutions, sufficient and insufficient evidence, long and short answers, or clear and unclear writing. Unfortunately, this frequently places the burden on students to disentangle the ambiguous, vague, and unspoken, yet structured language and logic of the discipline.  Students must often master these language-laden mathematical practices through trial and error, all the while developing conceptual understanding, evaluating arguments placed before them, and attempting to craft viable mathematical arguments of their own.

The Common Core State Standards describe mathematically proficient students as those who are capable of “breaking [situations] into cases,” and able to “build a logical progression of statements,” and “justify their conclusions [and] communicate them to others” (CCSS.Math.Practice.MP3).  Developing this student proficiency can be facilitated when teachers actively engage in the same types of systematic sensemaking they expect of their students.  By similarly inquiring into their practice as math teachers, these teacher researchers can break down mathematical practices, build pedagogical practices that purposefully attend to mathematical content and expectations held by the field, and justify and communicate their conclusions to students, colleagues, teacher educators, and other stakeholders.

This article traces one practitioner’s sensemaking around viable arguments from two perspectives–a high school mathematics teacher (Andrew Paulsen) and a university-based researcher (Meghan Odsliv Bratkovich).  To organize this discussion, the following sections each speak to one facet of knowledge for, in, and of practice (Cochran-Smith & Lytle, 1999, p. 250).  In the knowledge-for-practice section, I (Meghan, university researcher) break down the sensemaking process by which Andrew enhanced his understanding of mathematical argumentation.  In the knowledge-in-practice section, we speak jointly about how making sense of unnoticed aspects of mathematical reasoning reconstructed our understanding of content.  Building this more refined understanding of what mathematical reasoning involved catalyzed changes to Andrew’s teaching practice that purposefully attended to the previously-unseen language expectations central to determining argument viability.  In the final section, I (Andrew, practitioner researcher), communicate a vision of knowledge-of-practice that invites other teachers to engage in similar collaborative, multilayered sensemaking.

Sensing that his ninth-grade, Algebra I students seemed to ‘understand the math,’ yet struggle to craft arguments he could see as ‘effective,’ ‘clear,’ or ‘viable,’ Mr. Paulsen (as he is known to his students) began to question what he could do to help students meet Mathematical Practice 3.  Revisiting previous arguments that he used in class as exemplar models, I challenged him to figure out and name what it was specifically that made him view them as ‘effective,’ ‘clear,’ or ‘viable’, (as opposed to ‘right,’ ‘accurate,’ or ‘correct,’) in the first place.  This led to an ‘aha’ moment about what it means to ‘understand math’ that eventually led Mr. Paulsen to three takeaways: deeper awareness of his own expectations, new perspectives on his content, and growing appreciation of the importance of attending to how sense is constructed and conveyed in mathematics.

Mr. Paulsen had always held expectations for argument clarity.  However, he selected exemplars based on how convincing or moving they were to him and had never unpacked the components and criteria that led him to judge them as clear or unclear. He expected arguments to meet a tacit threshold for clarity, much like Supreme Court Justice Stewart, who famously did not attempt to define his threshold for obscenity, instead stating “I know it when I see it” (Jacobellis v. Ohio, 1964).  He essentially held up arguments with the hope that his students would be able to intuitively sense why they were clear—what made each exemplar exemplary.

To move beyond this, I encouraged Mr. Paulsen to apply similar systematic reasoning skills to those he teaches, aiming to explicitly define his de facto ‘know-it-when-I-see-it’ criteria for clarity.  Guided by little more than my simple questions such as ‘what about this part makes it clear?’ he identified discursive patterns between and within his exemplars.  He noticed how sentences built on one another and played specific roles within each argument, and for the first time realized his implicit reliance upon mathematical language structures when evaluating argument clarity.

Mr. Paulsen deduced that his exemplars often follow an answer-definition-example structure where a claim is made, a mathematical rule is stated, and an explanation linking claim to rule is offered.  This parallels the Toulmin ‘claim-evidence-warrant’ structure common to many disciplines, and recently adopted by other departments at Mr. Paulsen’s school.  He also found that exemplars were less reliant upon mathematical terminology like multiples, domain, and functionthan anticipated, and surprisingly, that some ‘everyday’ words (e.g., must, any, since, because) strongly influenced argument clarity.

This noticing of mathematical language, however, does not imply that expertise in linguistics or English Language Arts is necessary, though mathematics teachers often view language as outside their content (Harper & de Jong, 2009).  On the contrary, Mr. Paulsen’s sensemaking process highlights the sophisticated language expectations teachers already hold and reveals unseen expertise in the language of mathematics.  Many aspects of mathematics, such as the argumentation discussed here, simultaneously require discipline-specific structures, patterns, and word usage.  Therefore, socializing students into these mathematical discourse practices (Moschkovich, 2007) falls under the purview of mathematics teachers because it is not the teaching of language, but the teaching of mathematics.

Mr. Paulsen’s efforts invite other practitioners to undergo their own sensemaking process and seek their own ‘aha’ moments.  This process positions teachers to notice the overlooked language of mathematics, see their own language expertise, and reveal unseen facets of their existing content knowledge. This sensemaking empowers teachers to see themselves as those best equipped to sense, make sense of, and become sensitive to the language of mathematics.

Once Mr. Paulsen felt he had a better handle on how mathematical sense was made and constructed, we considered how this information could be used to give more meaningful instruction and formative assessment to his students.  Instead of planning to simply show another exemplar argument, we chose to create and juxtapose two different arguments for his students to discuss in class.  Using language inspired by multiple student work samples, we created two viable arguments for an adapted MARS Task (Toy Trains) his class had recently completed (see image below).

Maria says that a train set from Julio’s Puzzles has 40 wheels.  Can Maria be correct?



Maria is correct because Julio’s Puzzles only sells trains with exactly 8 wheels. In the real world, stores only sell full train sets.  This means that any train sets Julio sells must be sold in multiples of 8. In this example, Maria claims to have a train set with exactly 40 wheels. Since 40 is a multiple of 8, Maria is correct. Maria is correct that she says a train set from Julio’s Puzzles has 40 wheels.  Maria must buy a plethora of train sets. Trains from Julio’s Puzzles do not necessitate an engine, unlike Alex’s Toy Shop.  Therefore, Maria’s argument is affirmed, and she will be able to obtain a train set from Julio’s puzzles that has 40 wheels.

Mr. Paulsen’s goal was to draw on his own knowledge and sensemaking of mathematical argumentation to push his students to notice and focus on the overall structure and discourse-level features of a viable argument in mathematics rather than the word- and sentence-level features that both he and his students had focused on in the past.  Knowing that his students tended to regurgitate phrases in the question and fish for vocabulary, we crafted Argument B to use many of the “big words” and question re-phrasings that the students associated with good answers, but contain little logical structure.

We were careful to use the word mustin both arguments to show how a word can contribute to mathematical reasoning in one case (Argument A), but not another (Argument B), and was careful to make each argument of approximately equal length to avoid another common student misconception that better arguments were simply longer.  Argument A used more simplistic words, but included a claim, a statement of mathematical base case or universal truth, and how the specific case related to the base case.  In short, it used the internally cohesive structure consistent with a viable mathematical argument that Mr. Paulsen had previously identified.

The class on creating viable arguments opened with a Do Now we created in which the students were asked to reflect on what it means to “construct a viable argument” and to identify the structural components of a viable argument in English class.  This was intended to focus the students on reasoning and to link the structure of mathematical argumentation with a similar claim-evidence-warrant structure the students were familiar with from ELA.

Mr. Paulsen then presented the structure of a viable argument to the class, mapping his own sensemaking onto the claim-evidence-warrant structure used throughout the school.  Narrating a smartboard slide, Mr. Paulsen described a claim as the answer to the question, evidence as a universal truth, and a warrant to show how the specific case is an example of the evidence.  He then guided students through a sample argument, focusing specifically on the new and unfamiliar component of a universal truth by using a comfortable and familiar example–that all even numbers were divisible by two.

After leading a discussion about this straightforward example, students were then given the two Toy Train arguments we had created.  In addition to comparing and contrasting Argument A and B, the students were asked to identify the components of each argument and make a conjecture about which argument they thought a college student would use.

As expected, the students initially focused on specific words they associated with academic writing, such as the words ‘plethora,’ ‘necessitate,’ and affirmed,’ but the class soon realized that “fancy vocabulary” did not necessarily equate to a “viable argument.”  However, Mr. Paulsen did draw students’ attention to certain specific words, namely since, which signaled the base case was the reason that the specific case was true.  He emphasized how sinceshowed the causality between the evidence and warrant in a way that and, a transition word commonly used by the students, did not.

Importantly, though Mr. Paulsen provided a structure for argumentation and attended to specific words, he was careful not to turn the writing of mathematical arguments into a prescribed procedure.  He repeatedly pointed out multiple ways an idea could be expressed and challenged students to provide different word choices as the class co-constructed viable arguments.

By the end of the class, most students were able to choose Argument A as the more viable option, and identify the claim, base case (evidence), and specific case (warrant) within Argument A.  Thoughmany students still tended to describe Argument A using the same vague language often used by teachers (it needed to “be clearer” or “elaborate more”), some students were able to express specific ways that both Argument A and their own arguments could be improved to better mirror the structure and viability of Argument B.

What began as an attempt to create an intervention for my students to make sense of constructing viable arguments quickly revealed the need to better understand my own sensemaking.  My advice to students to “show your thinking,” “say how you got there,” or “be clear” did not provide sufficient scaffold for students to be able to craft a viable argument.

After Meghan asked me to reflect on what it means to ‘be clear,’ I recognized that I needed to dig deeper and unpack my own knowledge and expectations about the argumentation process that I had taken for granted in my practice.  I shifted from evaluating the arguments my students were producing to systematically reasoning through my own mathematical understanding and expectations for arguments.  By engaging in this process, I could see my own expectations more clearly and was able to articulate what I had been pushing my students to do.  I then worked with Meghan to develop a modified claim-evidence-warrant structure to scaffold arguments for my students and guide my student feedback to be more meaningful, specific, and actionable.

Though Meghan’s questions prompted this journey, her asking about clarity led me to my own questioning about mathematical reasoning. I am now more cognizant that my previous instruction was only sufficient for students who could adequately guess what I meant when I told them to “be clear.”  This process of inquiry helped me peel back the layers of mathematical expectations for argumentation and scaffold students beyond the sentence or vocab level so that they can make arguments at the discourse level.

The process of sensemaking was just as important as the product of the intervention. In other words, instead of jumping over the process (my inquiry) to get to the product (the intervention), it is just as important to focus on the sense-making process.  The intervention we produced met my intent to better prepare each of my students to access Mathematical Practice 3, but that was not the only valuable takeaway. As a teacher and lifelong learner, engaging in this inquiry not only prepared me to better understand one concept or teach one topic; it prepared me to go back and investigate any mathematical practice (such as what it means to “attend to precision”), which motivated me to share this sensemaking process with other teachers.  In the same way that Meghan’s inquiry kick-started the reflection on my teaching practice, I hope that this research can model and scaffold sensemaking in a way that can support other math teachers trying to come to a better sense of their own mathematical expectations, assumptions, and practices.


Cochran-Smith, M., & Lytle, S. (1999). Relationships of knowledge and practice: Teacher learning in communities. Review of Research in Education, 24, 249-306.

Moschkovich, J. N. (2007). Examining mathematical discourse practices. For The Learning of Mathematics, 27(1), 24-30.

Harper, C. A., & de Jong, E. J. (2009). English language teacher expertise: the elephant in the room. Language and Education,23(2), 137–151.

Jacobellis v. Ohio, 378 U.S. 184 (1964)

Reflections from the 2018 Agile Mind Academy

“Mathematics has the greatest and most indefensible differences in achievement and participation for students of different ethnicities, genders, and socioeconomic income levels of any subject taught in the United States.” Jo Boaler, Mathematical Mindsets

During this year’s Advisor Academy, Agile Mind brought together advisors that offer services for partners districts and used Mathematical Mindsets to ground our thinking for the week. The quoted text above was particularly eye-opening for me, and reinforced the importance of our work as we look to transform the teaching and learning of mathematics and science throughout our country. Vice President of Professional Services David Savage kicked-off the Academy with a session that made us think critically about the need to “double-up” our impact and pushed us to consider what we need to do to support more teachers and students. To help us achieve this bold vision, we use intense data-mining to help determine exactly where we need to go. As Agile Mind CEO Linda Chaput put it, if you want to build a garden, you need to watch where the people walk.


To that end, we engaged in critical conversations throughout the week about how we can continue to impact change in science and math classes everywhere. Linda asked us to reflect on the resources that are available for aspiring doctors, lawyers, and engineers as they learn their craft, and wondered why the same quality of materials are not available to pre-service teachers. While I have often talked about how Agile Mind was developed to help improve achievement for some of our most vulnerable students, our organization truly values educators and looks to empower teachers with high-quality resources to achieve this end.

During this inspirational and informative week, we also had a chance to learn about cutting-edge research from leaders at the Charles A. Dana Center. This year, we took a ‘deep-dive’ on the concept of ratio, rate, and proportional reasoning. Susan May, Lisa Brown, Susan May, and Kathi Cook helped explain a big mathematical shift from elementary school to middle school, as students move from working mostly with additive relationships (how much taller is John than Matthew?) towards multiplicative relationships. In the common core state standards, sixth grade is where this major change in thinking happens, but many math teachers take this important shift for granted (ask any middle school teacher what topic students struggle with the most, and they will undoubtedly say, “fractions”). These leaders explained that students need to develop an understanding of ratios without using a fraction notation that can cause confusion, which was a major “ah-ha” moment for many of us.

We also had a chance to think critically about how to unleash all students’ power through differentiation with Abby Neumeyer. I believe now more than ever that we need to re-think our connotation of what differentiation really means and looks like in classrooms. Traditionally, many teachers have thought of differentiation as offering different tasks to different students. While this may be an effective method in certain limited situation, Abby encouraged us to instead re-think our definitions of what effective differentiation really is, and pushed us to ensure that we are upholding high expectations for every student.

I also cannot thank Abby enough for her dedication to make a professional services hub that will help empower our advisors moving forward.

Joyce Boubel leading a session on serving emergent biliguals

Joyce Boubel led a thought-provoking session which highlighted that our public schools are becoming increasingly more diverse and more students than ever are emergent bilinguals. As such, it is crucial that we consider how to best support these students as they progress throughout their formative years. In collaboration with the Dana Center, Agile Mind has published research-informed best practices to best reach students learning the English Language while upholding high academic expectations for all students. Joyce shared many of these best practices, and feel free to read a summated version of this resource here.

Massie McAdoo also led an informative session on formative assessments, and highlighted this passage from Mathematical Mindsets: “[Paul Black and Dylan Williams] found something amazing: a form of assessment so powerful that if teachers shifted their practices and used it, it would raise the achievement of a country, as measured in international studies, from the middle of the pack to a place in the top five… if teachers were to use what is now called ‘assessment for learning,’  the positive impact would be far greater than that of other educational initiatives such as reductions in class size.”

Massie shared how every Agile Mind course program encompasses thousands of formative assessment items, many of which can be automatically scored and reported in real time, thus providing instantaneous, targeted feedback for learners while enabling teachers to use data to impact instructional decisions in the moment. One of the great things about working at Agile Mind is that we are always talking about scale. In fact, “to date, almost 4 million students have engaged more than 3 billion times with our embedded assessment tasks. Data on their experiences are used to enhance tasks to ensure they fulfill the promise made by assessments: that if learners have mastered the necessary knowledge, they will succeed.”

Talking with Susan May from the Dana Center

Although this was my second time attending the Advisor Academy, it was my first time attending as a Senior Advisor. I was excited to share some ideas I had about leveraging social media to increase our outreach with school leaders, facilitate an NCTM-inspired ‘Ignite!’ session, and lead a workshop on best practices from the research of adult development theory; I can only hope that Dr. Drago-Severson would be proud!

Overall, it was truly an inspirational week with such a talented group of educators that are making a difference in schools throughout the country. After a busy two months, I honestly could not be prouder to be part of such an incredible team and I look forward to working and learning with them as we continue our critical work in improving our public schools throughout the country.

Exploring Cambodia, the Kingdom of Wonder

Cambodia was one of my favorite countries that I have ever been privileged to visit. I started my trip last month in Siem Reap, a small city in the middle of the country that has hundreds of well-preserved temples that are thousands of years old. Thousands of years ago, the city was called Angkor, a megacity that contained 0.1% of the world’s population at its heyday. Angkor served as the capital of the mighty Khmer Empire, an empire that was a powerful state in South East Asia. The kingdom was so large, in fact, that between the 9thand 15th centuries, the Khmer Empire covered much of what today is Cambodia, Thailand, Vietnam, and Laos.

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Standing in front of the Royal Palace in Phnom Penh

Exploring Angkor Wat, Bayon, and other structures around Temple Town was truly an indescribable experience. After all, “Angkor evokes the image of a giant palimpsest – like a medieval parchment erased and overwritten repeatedly. The capitals of the Khmer Empire were established here one after another, one on top of the other, almost continuously for over seven centuries, profoundly shaping the entire landscape. Over time, a dense and complex matrix was created, evidenced primarily by the spectacular succession of monumental temples. These grandiose structures made of brick and sandstone have survived the passage of time, unlike the greater part of the cities whose houses and palaces, built of wood and other perishable materials, have disappeared into the jungle leaving behind the skeletal remains of the Angkor civilizations seen today. Nonetheless, the remaining skeleton is so impressive that naturally everyone tries to imagine the temples at the time of their splendor, bustling with life and color” (Angkor Guide Book).

Angkor slowly started losing its influence during the 15th century. Although many historians still wonder what exactly brought down the mighty Khmer Empire, we know now that the neighboring Siamese kingdom in Ayutthaya (modern-day Thailand) conquered the kingdom during the mid-1400’s. The Kingdom of Angkor slowly drifted into irrelevance, until the French colonized “Indochina” in the mid-19th century. Cambodia was placed under French rule for almost a hundred years, until Prince Norodom Sihanouk (who claimed Angkor King lineage) won independence for his country. The United States, upset over his rise to power, helped Lon Nol set up a successive military coup, which led to a brutal capitalist form of government. Many Cambodians hated the US-backed Nol and supported an up-and-coming rebel group that called themselves the Communist Party of Kampuchea (CPK), otherwise known as the Khmer Rouge.

Under the brutal leadership of “Brother Number One” Pol Pot, the Khmer Rouge took control of Cambodia on April 17th, 1975, immediately renamed the country Democratic Kampuchea, and started a brutal genocide that tortured and killed anyone with alleged ties to the former government. By 1980, the crimes of the Khmer Rouge were well-known: Pol Pot ordered the execution of two million of his own people (about one-quarter of the population), including 80% of teachers and 95% of doctors. Cambodia, as Pol Pot liked to say, was returned to “year zero.”

During my time in Cambodia, I humbly visited one of the mass killing fields and “Security Prison 21″ (S-21), also known as Tuol Sleng, “a former high school in central Phnom Penh that the Khmer Rouge had turned into a torture and death house. Vietnamese journalist and soldiers first walked in to find rotting corpses still in shackles. Gruesome torture implements lay about, and in an outbuilding the soldiers found thousands of pages of records, including photographs of the victims. At least 15,000 Cambodians had been taken there, tortured until they confessed to being an enemy agent, and then killed – whacked on the head with an iron pole” (Cambodia’s Curse).

Unfortunately, Pol Pot was successful in returning Cambodia to year zero, and the country has never fully recovered from its dark past: By the tenth grade, 87% of students have dropped out of school. In fact, most Cambodians today over the age of 35 have had little formal education; the average family makes about $3.00 USD a day, and Cambodia remains one of the poorest countries in the world. Visiting this high school turned death camp turned genocide museum is an important reminder to all of us as to what can happen when we allow a narcissistic autocrat to separate families, make false promises, scapegoat immigrants and political opponents, lie to their citizens, and make a free press the “enemy of the people” (as a side note, we are often mortified when we look at the horrors of the past, but too many of us remain silent about the injustices being carried out in the United States of America right now).

Yet in the midst of this deep oppression, I somehow found myself at the most special place I have ever visited in my entire life: The LaValla School. Amongst all the devastating poverty and oppression that has long crippled the Cambodian people, lies one of God’s greatest treasures, where you can hear the laughter and feel the joy of the children from beyond the gates. The LaValla School is (somehow) the only government-approved school in the entire country of Cambodia that provides a full primary education to children with physical disabilities. Founded by the Marist Brothers over twenty years ago, the amount of genuine joy and happiness at this place is truly remarkable.

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The second I arrived at LaValla, students came running up to me and gave me a big welcome hug. One of the students took my hand and started showing me around the school right away; I have never felt so welcomed during any school visit in my life. To be fair, this place is more than just a school – it is a true community.

The Kingdom of Cambodia truly stole my heart this week. After decades of genocide, corruption, oppression, and extreme poverty, the Cambodians that I met were some of the most resilient and welcoming people that I have ever known. That being said, all of us are “guilty of romanticizing squalor and imagining that a simpler life must be happier and more content than our own. It is simply not true when the measure of that simplicity is no access to medicine, education or clean food and water . . . [yet] there is no denying the subtle charms of lives lived with a careful attention to the present and the everyday” (Walter Mason).

On the day of my flight back to Kaohsiung, I never cried so hard leaving a place in my entire life. Perhaps Mike Duffy, who visited the Lavalla School for two months last year, put it best: “I have obviously struggled to put into words the soulful beauty of life here, where God is ever so vibrantly present. It is something that I will do my very best to continue to share with you all, as this experience has already changed the man I am. With a filled heart and tremendous pleasure, I am adding some pictures to try to encapture LaValla, the children, and their joy to be alive.”

Thank you for a truly transformational week, Kampuchea! This school, these amazing children, and this beautiful country will be in my heart forever.


An American Abroad

This past memorial day, on my flight from Kuala Lumpur back to Kaohsiung, I had the privilege of sitting next to an active-duty member of our U.S. military currently stationed in Okinawa, Japan. During our four-hour flight, we had an incredible conversation that covered just about every topic, including the role of America’s military in 2018, the current geo-political climate of the world, and yes, Mr. Trump. The marine I sat next to told humorous, informative, and enlightening stories about being stationed in Romania, training troops in Bulgaria, and moving to Japan. Towards the end of the flight, I asked him what his current responsibility in the military was, and he talked about how, in his perspective, the U.S. military’s greatest strength is in soft diplomacy. He talked about the need for our soldiers to gain trust in foreign lands by getting to know the locals, understand their culture, and grow to mutually understand one another. Although we were the same age and had taken radically different paths in life, we both somehow ended up on the same flight, headed to the same destination, with the same goal in mind: to advance the virtues of the United States through the use of soft power.

Decades ago, Joseph Nye originally coined this term – soft power – in his critically-acclaimed work Bound to Lead: The Changing Nature of American Power. He wrote that, “if a state can make its power seem legitimate in the eyes of others, it will encounter less resistance to its wishes. If its culture and ideology are attractive, others will more willingly follow… when one country gets other countries to want what it wants, [it] might be called soft power, in contrast with the hard or command power of ordering others to do what it wants.” More recently, Nye has suggested that the “best propaganda is not propaganda,” and that “credibility is the scarcest resource.” Former Secretary of Defense Robert Gates believed in this vision, and advocated that the Bush administration consider “a dramatic increase in spending on the civilian instruments of national security – diplomacy, strategic communications, foreign assistance, civic action, and economic reconstruction and development.”

Not a day passed by in these last eight months that I did not work hard to advance the soft power efforts of our country. As a cultural ambassador for the United States, I was invited to meet Taiwanese President Tsai Ing-wen, had an interview with Taiwan’s Minister of Foreign Affairs Joseph Wu, and gave a presentation to the U.S. Assistant Secretary of State Marie Royce. On a daily basis, I had the opportunity to interact with ordinary people that have never met an American before. I visited at least one high school (in Tainan) that told me that I was the first American to ever visit their school. On many occasions, I was asked representative questions about the United States, and while I always strived to be welcoming and open to dialogue, I always made it clear that I was sharing my opinion, as there is no such thing as a representative American perspective. In truth, I was trying my hardest to avoid the danger of telling a single story.

I also had the opportunity to visit the cold war outpost of Kinmen and even played basketball with students that are living in a former warzone – talk about true soft diplomacy! Kinmen (formerly known as Quemoy) was the site of the last battles of the Chinese civil war, and was later shelled heavily by the PRC’s Liberation Army throughout the 1950’s. The Taiwanese island is so close to the mainland that my cell phone network automatically switched to a mainland Chinese phone carrier! This trip also served as an important reminder of the urgency to study our history so that we may never repeat the mistakes of our past. For me, I still question how I somehow have made it through 13 years of public education (and an additional 8 years of university) without every hearing about the role Quemoy and Matsu played in the Taiwanese Strait Crisis.

Speaking of learning about our past, I also saw the atrocities of war first-hand when I visited Vietnam. Unlike many conflicts, including World War II, the Vietnam War ended in the 1970’s. This means there are millions of people still alive that remember the war like it was yesterday, and one cannot walk a single block in Ho Chi Minh without seeing someone that has been negatively impacted by Agent Orange. It was sad standing at the War Remnants Museum, next to captured American tanks and airplanes, and asking whether all these deaths, destruction, and birth defects were worth it. At the time, many Americans subscribed to the so-called “domino theory,” and believed that if one country were to become communist, all of Asia would quickly become communist. In hindsight, it is clear just how wrong many of those leaders were in our past. In some ways, America in 2018 in similar to America in 1968, because during the war, “the United States was widely unpopular around the world, as it is now… Yet despite unpopular government policies, our openness and self-criticism allowed the American idea to retain its appeal. A free press, independent courts, and a Congress willing to confront the executive branch can provide a similar measure of soft power today.”

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Standing at the War Remnants Museum in Ho Chi Minh City next to a U.S. Army helicopter captured during the Vietnam War.

It frustrates me to no end when people accuse me of being anti-military because I am anti-war. I have the deepest respect for our armed forces and will be forever grateful for their sacrifices. But we must question, why do we constantly get involved in meaningless conflicts that put our brave soldiers in harm’s way to begin with in the first place? My grandfather, who meant the world to me, served in the Korean War. While we only have a few photos of him during this time period, look at how happy he looks when he is with his fellow Koreans. I particularly love the photos of him standing with Korean children. To me, I have no doubt about it: my grandfather fundamentally believed in the idea of soft power, and worked hard to win over the hearts and minds of the Korean people.

In fact, I was the first person in my family to visit the Korean peninsula since he was honorably discharged almost 70 years ago. Standing in the de-militarized zone and briefly walking into North Korea was an experience that brought many personal feelings to bare. I can only hope that he would be proud of me.

Nothing, however, was as powerful as visiting Hiroshima.

At 8:15am on August 6th, 1945, a U.S. army B-29 bomber dropped an atomic bomb on the Japanese city of Hiroshima, and our world was changed forever. It was surreal being one of the few Americans to ever walk around the atomic bomb dome and the Peace Memorial Museum, and I was at a particular loss for words when I was standing directly underneath the hypocenter of the bomb. When visiting the memorial, it was interesting that the memorial did not solely blame the United States, but rather both countries, for putting their egos in front of the lives of innocent citizens. For me, the hardest part of the day was visiting an elementary school that survived the initial impact, where a majority of the young students were killed instantly.

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“I heard what others have said about Hiroshima. Now I have seen for myself, and I am utterly devastated.” -Desmond Tutu

While visiting Hiroshima was without question one of the most difficult places that I have ever been to, there were many difficult conversations I had throughout my time in Asia. When I met with Assistant Secretary of State Marie Royce, I told her how difficult it was being asked questions about the current administration or our country’s absurd obsession with guns everywhere I went (I encourage all of my fellow Americans to read an op-ed I wrote that boldly declares that our problem with gun violence is #NotNormal). Truth be told, I was asked questions about school shootings and Trump on a daily basis during my time abroad, which were unequivocally the hardest to answer as a representative of our country. It is not necessarily because of Trump’s policies, as I am all for a political disagreement – dissent and diversity of thought and culture is what truly makes America great. Truth be told, it has been embarrassing to be an American abroad during the so-called era of Trump. Simply put, our president does not represent our country well, and respect for America around the world is in an objective freefall (well, except for in two countries: Russia and Israel). As I told our Assistant Secretary last month, it has been hard serving as a cultural ambassador for the United States during this tumultuous time period under an administration that does not believe in soft power.

Why has it been so hard answering question about Trump? Maybe it is because he praises dictators while he is incredibly hostile with our closest allies. He openly mocks disabled people, and makes lewd remarks of women. His Secretary of Education knows nothing about schools while his EPA Director has repeatedly abused his authority. After complaining for years about how much golf President Obama has played, he has played more golf than any other president. Trump and his administration lie about everything, including paying off people that he had had affairs with and meetings during his campaign. He is a proponent of separating vulnerable families at the border and wants to widen the death penalty for drug dealers. He calls African nations “sh*t-hole countries”   and calls immigrants of color, “animals.” And to put the figurative sprinkles on top, he neither believes in the power of soft diplomacy nor the mission of the Fulbright Program.

On the contrary, I have seen the innate power of soft diplomacy first hand. I have gotten to know so many amazing people, in a part of the world that is greatly misunderstood by the west. I have come to have a deep appreciation for Taiwanese hospitality, and the importance of standing up for what is right.

At the end of the day, the “greatest threat to the American idea is what we may do to it ourselves. Terrorism is like jujitsu: The small players win if they make the large player use his strength against himself. If we respond to terrorism by becoming less open—economically, socially, and politically—we lose. As George Kennan warned in 1946, at the start of the Cold War, the greatest danger that can befall us is that we shall allow ourselves to become like those with whom we are coping.”

Here is my charge to my fellow Americans: We can do better. Scratch that, we must do better. I encourage you all to get your information and news from objective news sources, such as BBC and NPR, and stop watching outlets such as MSNBC, CNN, or FOX. If you have the opportunity, travel abroad as much as possible, and strive to really get to know the people of foreign countries and their culture and way of life. Try to listen to others more, especially people that do not look like you or may have different worldviews. Push your thinking by watching documentaries and reading books and attending local lectures and talks. Perhaps most importantly, reflect critically on your belief system, and constantly ask yourself why you believe what you believe. If there is anything that I have learned in my first 27 years of life, it is that people are people everywhere. At the end of the day, everyone I have ever met wants their family to be healthy, their kids to be successful, and to be treated fairly and justly. Is that too much to ask?

When President Obama visited Hiroshima, he reminded us that, “we have long known the agony of war. Let us now find the courage, together, to spread peace and pursue a [better] world.” Perhaps I am an eternal optimist (and pacifist), but I fundamentally believe that we can make this world a better place if we try. I mean, why not aim to create good instead of evil? Why not look to make peace instead of war? Why not try to spread love and compassion instead of hate and arrogance?

Now (especially now) is the time for an effort like never before to achieve peace.

Let us all take hands.

This post is dedicated to Anthony Bourdain, someone that knew how to travel and celebrate life better than anyone. He worked relentlessly to show us that we should not be afraid of strangers in our global community, and that people are people everywhere. Thank you for inspiring so many people (including myself) to take this world by storm, Mr. Bourdain. 

Taiwan truly is the heart of Asia.

A bit of madness is key
to give us new colors to see.
Who knows where it will lead us?
And that’s why they need us!
So, bring on the rebels,
the ripples from pebbles;
the painters, and poets, and plays.
And here’s to the fools who dream…
crazy as they may seem.
Here’s to the hearts that break…
here’s to the mess we make.

It was a hectic couple of days leading up to my departure for this incredible journey. After a snow day, an emotional last day at East Side, and a truly surprising going-away party (thank you mom, dad, and Kim!), I was off to Asia to embark on the opportunity of a lifetime. I was super excited about visiting Japan and moving to Taiwan, but I also became sad about leaving home (for really the first time in my life).

As I sat on the plane, I reflected on leaving everything and everyone that I knew behind. I started to realize exactly what I had gotten myself into, as I was about to move to a country where I did not speak the language, personally knew nobody, and could not even read a menu. After enjoying the in-flight service and reading for a while, I decided to watch a movie, and happened upon La La Land, a movie that claimed the Oscar for Best Picture last year (well, for two minutes, at least). I felt that the beginning of the movie went slowly, and the only reason I did not turn it off was that I was stuck in an aluminum tube for the next twelve hours. I quickly fell in love with La La Land, as it intimately shows the often-untold sacrifices that people need to go through to chase their dreams. And that was before the audition scene:

In a weird way, it was almost like Emma Stone was speaking to me. Other than perhaps Stand By Me, rarely has a Hollywood Blockbuster ever had such a profound impact. I genuinely do believe that traveling is essential “to give us new colors to see,” as Ms. Stone so eloquently sings in this scene. Perhaps most importantly, she reminds us the importance of always striving to approach new situations with an open mind, as “who knows where it will lead us?” Watching La La Land reassured me that everything was going to be alright, and I was ready to take on this world by storm!

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At the Fo Guang Shan (佛光山) Buddhist Monastery in Kaohsiung.

After landing at Narita airport right outside Tokyo, I was excited to visit Japan and see Asia for the first time. On my first morning, I got up at 3am to get in line for the Michellin-starred SushiDai restaurant, which was hands-down the best sushi I have ever had in my entire life. After visiting four major cities (Tokyo, Kyoto, Nara, and Hiroshima) and falling in love with Japanese culture and their way of life, I was off to my Fulbright destination: Kaohsiung City, Taiwan.

My first month in Taiwan was really busy, as I had a lot of logistical things to take care of, including securing my apartment, completing all of my immigration paperwork, and adjusting to my new life for the next six months. After meeting my advisor, new colleagues at Kaohsiung Girls High School, and grabbing lunch with some former Fulbrighters, I was off to Taipei to speak at the Fulbright mid-year conference. After touring around the capital of Taiwan for a week, I was so honored to be invited to celebrate Chinese New Year with Michele’s family. During the holiday, all of the schools throughout Taiwan shutdown, and many of my friends took advantage of the extended break to travel internationally. February was definitely the hardest month to get through. Although I had the amazing opportunity to attend the Olympics and watch our women’s hockey team beat Canada to earn a gold medal, there were a lot of tough transitions to get through along the way.

You see, living abroad for an extended amount of time is an incredible experience, but it is not always rainbows and unicorns as Instagram would make us believe. As they told us during our Fulbright orientation, culture shock is a real thing, and there are times when it can be very lonely traveling by yourself (although I always try to make the best of every opportunity). The time change was difficult working through, too, as I almost always had to FaceTime with my friends and family in the morning before they went to sleep. Traveling to Asia also emulated what it must be like to be illiterate, as few signs (outside of airports and subway stations) are in English. On more than one occasion, I ordered something random from a restaurant with no English menu (which was almost always delicious, for the record, but is still an anxiety-causing experience). I may have learned 一點中文 , 我不台外說中文 , but I can still only read about 40 or 50 Chinese characters.

Once March rolled around, I had a bunch of school visits lined up and ready to go. I have learned so much about Taiwanese culture and have grown to love many aspects of this special country. I will definitely miss all of the incredible hospitality, amazing food, and ‘futuristic’ transportation options. I spent every Monday learning from the incredible educators at Kaohsiung Girls High School and enjoyed our weekly “Subway” meetings. I have come to really look forward to every Monday, as we had the chance to talk about education, politics, and yes, even gun violence (a future blog post will consider my thoughts about being abroad during this tumultuous time period in American history). On a serious note, I really cannot thank Hsin-Wei Chen [陳欣薇], Yi-Fan Chen [陳怡帆], Yi-Hsiang Chang Chien [張簡逸翔], Jia-Wei Sun [孫嘉偉], Hsun-Hsun Chung [鍾恂恂], and Hsien-Tsung Lin [林憲聰] enough for everything they have helped me with. 謝謝!

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With this awesome crew during a “Subway Meeting” after an observation at KGHS

During the Spring, I had the chance to speak at a plethora of conferences and visit a few other countries, as well. The longer I was in Taiwan, the larger my network grew, and I was soon invited to visit schools in every major city. I really appreciate Mr. Chen [陳光鴻], who is leading the effort to implement Phenomenon-Based Learning throughout Taiwan. I held focus groups in Taichung, taught math lessons in Kaohsiung, and visited vocational schools in Tainan. I watched students riding unicycles in Kinmen, and met Taiwanese President Tsai ing-wen in Taipei. Over time, I slowly acclimated to Taiwanese culture (and yes, I had plenty of amazing Taiwanese food at all of the famous night markets throughout the country).


Taiwan is special for countless reasons, but my favorite part can be summarized in three words: mindfulness of others. Although most social norms around personal space can be quite different in Taiwan, people often act with others in mind instead of only thinking about themselves. Take something simple, like letting people pass on an elevator: in the U.S., there are signs everywhere that tell people to stand to the right and let people pass to the left, but people still do not listen. In Taiwan, it seems as though everyone automatically does the right thing to make life easier for everyone else, and it is like that for everything. You can take a nap on a crowded MRT, because people are quiet. Once on the subway, people never sit in the designated priorty seat to make it easier for others that really need the seat (see photo below). The culture of respect creates an unbelievably safe atmosphere that leads to modern day miracles. On countless occassions, I have left my phone, wallet, and Macbook out at a busy Starbucks  and no one ever stole it. I have seen people leave their key in the ignition of their scooter, because they have such a deep trust that no one will steal it. When you cultivate a society that is mindful of others, life is better for everyone (even the dogs are rarely leashed, and are always well-trained). We have a lot to learn in the United States, where we value individualism over just about everything else, almost to a fault.

As I wrap up my time here in the Republic of China, I am often asked what I am going to tell my friends and family about Taiwan when I get back to the states. The first thing that I am going to talk about is the amazing Taiwanese hospitality. In the almost 30 countries that I have been privileged to visit, the people of Taiwan are some of the nicest and most welcoming people that I have ever met in my entire life. On a daily basis in Kaohsiung, people will come up to me and say ‘hello,’ and try to help me out in anyway that they can. Sure, the Taroko Gorge was one of the most beautiful parts of nature that I have ever seen, but the true beauty of Taiwan is its people.

Taroko - 1.jpg
The Taroko Gorge (太魯閣) is gorgeous, but the true beauty of Taiwan is its people.

Second, albeit controversially, is that I believe now more than ever that Taiwan is unequivocally a sovereign state that deserves the recognition and protection of an independent nation unconditionally. If there is any policy that I agree with the current administration on, it is their support of the Republic of China. Taiwan is a beacon of hope that shares the democratic values of the United States, and we need to ensure that our countries have an enduring friendship that will stand up to geopolitical rivals. Perhaps Director Moy, the de facto United States ambassador to Taiwan, said it best during our end-of-year Fulbright banquet:

In a weird way, my Fulbright experience is like a movie playing before my eyes. On some days, it feels like I have been living in Asia for years. On other days, I feel like it was just yesterday that I was sitting on a plane waiting to take on the world. As they say, maybe “a bit of madness is key, to give us new colors to see. Who knows where it will lead us? And that’s why they need us! So, bring on the rebels, the ripples from pebbles; the painters, and poets, and plays. And here’s to the fools who dream… crazy as they may seem. Here’s to the hearts that break… here’s to the mess we make.”

謝謝 , 台灣![Xièxiè, Táiwān!]

很高興認識你們 , 我學到了很多東西。[Hěn gāoxìng rènshí nǐmen, wǒ xué dàole hěnduō dōngxī.]

我會想你的![Wǒ huì xiǎng nǐ de!]

台灣將永遠在我心中。[Táiwān jiāng yǒngyuǎn zài wǒ xīnzhōng.]


The bǔxíbān (補習班) experience: My night at a Taiwanese Cram School

As previously mentioned, the Taiwanese educational system encompasses a gigantic range of cram schools. However, it is virtually impossible to find stories in the mainstream media about east Asian education today that boldly confronts the existence of this parallel educational system (Turton, 2012). These cram schools (known in Mandarin as 補習班or bǔxíbān, which literally translates to ‘supplementary learning class’) are akin to large tutoring centers that lecture students about mathematics, science, Chinese, and English.

Cram School - 1
A typical bǔxíbān in Kaohsiung, Taiwan

After hearing about cram schools in dozens of interviews conducted throughout Taiwan, I was visit a bǔxíbān and actually see what one is like for myself. Thanks to some of my colleagues at Kaohsiung Girls High School, I was able to visit one of the most famous cram school teachers, Teacher Lin, at 志光公職補習班 (Zhiguang Public Cram School). When I first walked into the cram school, I was surprised at how elaborate the facilities are. I really enjoyed how this school celebrated academic success and had a “Hall of Fame: of all the university acceptances form previous students. This learning center had several floors and different rooms for different needs: The main classroom (four different rooms for different subjects), a study room with a “teaching assistant” (but is a full-time licensed teacher), a computer room, and a “relax” room where kids can get water or tea. The classroom I visited was about twice size of a typical American classroom and had over 100 seats, but somehow does not feel crowded.

Cram School - 6

One of the most interesting aspects of the cram school structure is that all lessons are recorded for kids that are absent or were confused with a particular topic and wanted “extra help.” The school has a library of videos that teach students any topic imaginable and is akin to an old-school version of Khan Academy (with their current teacher). The students can watch the videos of the lesson anytime they want and can rewind or slow down the video at their own discretion at any time.

Cram School - 3

During the class, I was interested in how engaged the students were in the lesson. Teacher Lin has created his own resource (books) for each math topic, and each problem has a similar problem with a given answer for more practice. The use of multiple representations is rampant and is very strategic about using different colors to highlight how different parts of the pictorial representation relate to the procedural applications. Another interesting tidbit is that the teacher uses different parts of the board for different parts of the lesson. The far-left part of the board is kept up and used as a defacto anchor chart Teacher encourages different methods to solve the same problem and uses jokes and brain-breaks to divvy up the lesson. Perhaps most importantly, kids can still be kids – eat, drink, get up and take a break or use the restroom, etc.


After class ended, I had a chance to have an impromptu focus group with the students in the class, which was an interesting learning moment, too. Class got out at 9:30pm, and even I was tired – but I was not attending school all day! I asked the students what a typical day looked like for them, and they shared that their public schools get out at 5, cram school starts at 6 and gets out at 9:30, with a 20-minute break at 7:50. Students get home at 10, take a shower, relax, and start HW/studying at around 11. Students spend 1-2 hours and then go to bed around midnight. I also asked if any of the students hang out with their friends or play video games, to which they replied: “Yea! On the weekends mostly. And sometimes during school (haha)!”

Cram School - 8

I also asked the students their perspective about the Taiwanese education system, and they shared that, “we wish there was more collaboration… I don’t like sitting and listening to a teacher for three hours. I went to school in Denmark for a year, and it was very different.”

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Post-bǔxíbān dinner at Smokey Joes!

While the practice of late-night tutoring remains controversial, I was really impressed with some of the aspects that I saw during my bǔxíbān visit. It should be noted, too, that although many public schools in Taiwan are remarkably equitable, these night classes are one way in which parents with means use their resources to give an unfair advantage to their children, but often at an extraordinary financial and emotional cost (Williams, 2017). While visiting a cram school was a tremendous experience and I gained a lot of interesting ideas about pedagogy, I am deeply conflicted about pushing our students too hard. I am all for high expectations, but to a point. After all, shouldn’t kids be kids?


Works Cited

Turton, M. (2012). The View From Taiwan. Retrieved March 04, 2018, from http://michaelturton.blogspot.com/2012/03/friedman-on-taiwan.html

Williams, C. (2017). Teaching English in East Asia: A Teachers Guide to Chinese, Japanese, and Korean Learners. Singapore: Springer imprint is published by Springer Nature.

Fulbright Research Presentation

I am offically at the ‘T-minus one month’ mark, and I honestly cannot believe how fast this entire Fulbright experience has gone.

This past week, I had the chance to talk with Marie Royce (Assistant Secretary of State for Educational and Cultural Affairs) on the importance of soft diplomacy and how the United States needs to support the Fulbright Program now more than ever. I also had the opportunity to present my research at the Foundation for Scholarly Exchange a few weeks ago in Taipei. Below is a copy of what was printed on my poster that was created by the Fulbright Taiwan staff:

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The poster that was made for my final presentation

“Reflecting on hundreds of classroom observations throughout Taiwan, Paulsen will share his thoughts on the best practices of Taiwanese teaching techniques, education policy, and culture, and offer recommendations on how to implement these ideals in urban schools throughout the United States.

Andrew Paulsen is currently the Lead Math Teacher and an Instruction Coach at East Side High School, the largest comprehensive high school in Newark, New Jersey. Originally from Levittown, New York, Andrew received his B.A. from Marist College, his Master’s in educational leadership, management, & policy from Seton Hall University, and his Ed.M. in public school leadership from Columbia University’s Teachers College. He aspires to change the world, one student at a time.”

Here is a video of my final presentation:

After the presentation, I was presented a certificate from Dr. Vocke, the Director of Fulbright Taiwan.

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Receiving my Fulbright certificate

I am officially back in Kaohsiung for my last four weeks in Taiwan. After departing from this incredible country on July 15th, I am looking forward to traveling through mainland China with my parents for a few weeks, and then coming back to the United States at the beginning of August. Here is to one more awesome month in beautiful Taiwan!

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My Fulbright Taiwan Certificate of Achievement. I loved that they printed my Chinese name (柏安尚) on it, too!