In the Mood for Advice? Building Confidence with the Present Subjunctive

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FLHS World Languages Team: Greta Gieseke, Hillary Biles, Kathy Grant, T. J. Heupel, Alex Klenz, Suzanne Lea, Rebecca Wittig

One of the challenges in language teaching is managing expectations for each of the modalities (speaking, listening, reading, writing). As students progress from novice to expert in a second language, teachers work to appropriately increase expectations for independent use of the language and establish incremental milestones students should reach on the road to proficiency.

Listening and speaking are often the most intimidating modalities for students. Students hear words in the target language and the then have to "see" these words in their mind before they can attempt to interpret meaning. Unfortunately, students often give up before reaching this stage of development. For example, many hear Spanish and quickly stop listening if they don’t immediately understand.

Our world languages team has been tackling this problem for several years. Most recently, the team planned a research lesson for a Spanish 3 class focused on giving and receiving advice while using the present subjunctive.

What is the Present Subjunctive?

It’s all about “mood.” Up until this time, students learn about verb tenses. They learn how verb-endings in Spanish reveal WHO does something and WHEN the action occurs. With the subjunctive, the verb also indicates the MOOD of the action. This is common in Spanish but it’s difficult for English speakers to grasp, because the subjunctive plays a less prominent role in English. It appears in more formal contexts when we use verbs or expressions that suggest something is important or recommended. (e.g., It is vital that they be warned of the consequences).

In Spanish, the subjunctive is formed by changing the ending to the verb stem. For example the present indicative for “I drink” is “yo bebo.” The present subjunctive is formed by taking the stem (beb-) and adding a new ending (beba). *The rules for these endings change depending on the verb type and the subject (1st, 2nd, 3rd person, etc).

Since the subjunctive is used for recommendations and suggestions it’s particularly useful for giving and receiving advice. For example:

Mi doctor recomienda que beba más agua.

My doctor recommends I drink more water.

Matchmaking and Giving Advice

To help students gain confidence with giving and receiving advice in the subjunctive, the FL team designed the following “storyline” of lesson activities:

  1. Warm-up: pre-assessment of confidence level

  2. Identifying logical/ illogical advice (bueno/malo) - Noting the use of subjunctive

  3. “Matchmaker” speaking activity: match advice with a problem by speaking in Spanish.

  4. Slideshow activity: Provide various examples of media presenting problems. Ask students to complete a phrase, summarize a scenario, or give advice using whiteboards.

  5. Collect evidence: post-assessment

The teachers anticipated the “matchmaker activity” (segment #3 above) would be the pivotal moment of this lesson. At this stage, students were asked to speak in Spanish and match advice with problems without independently generating language. The segment was designed to build students’ confidence and scaffold their proficiency with advice-giving phrases and grammar (the subjunctive). This prepared them for the final “slide-show activity” where students demonstrated comprehension by summarizing scenarios and generated their own language for advice based on new problems presented in the slides.

Click here to watch a series of video excerpts from lesson segments #2 (bueno/malo), #3 (matchmaker), and #4 (whiteboard activity with slideshow).

Student Results

At the conclusion of the lesson, all three students we interviewed and at least half of the students from the pre-post assessment reported greater confidence in communicating advice.

Click here to watch interview responses from several Spanish 3 students.

Students also performed better than anticipated on the lesson assessment, giving reasonable answers for fill-in-the-blank questions (90%), correctly summarizing scenarios read in Spanish (70%), and providing advice in the subjunctive after listening to a problem in Spanish (80%).  (See evidence tally below.)

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Lesson Reflections

Nurturing Student Confidence. The students had several opportunities to explore the language and experience success in this lesson. This helped boost their confidence and prepared them for each successive activity with increasing cognitive demand and use of the target language. At the same time, addressing student confidence in listening is a process. Students will need continued opportunities to experience success over time.

Improving Listening Comprehension. Throughout this lesson, students had multiple opportunities to practice listening for subjunctive in low risk situations, reducing the affective filter. Reading new vocabulary and phrases in advance enabled students to better visualize those words during listening activities. Students also benefited from practice with summarizing and interpreting. In the words of one student, this helped them “not to read word for word, but for overall summary” and to grasp the gist of each scenario. Even if they could not catch every word, they could at least identify some details and not feel completely lost.

Responding in Spanish. In their written responses, most students demonstrated good understanding of the problems presented and provided sensible advice using the subjunctive. They made tangible progress in listening to, summarizing, and writing down advice in Spanish. Speaking, on the other hand, remained a challenge for many students. During the matching activity, for example, several students held up their slips of paper for others to read, rather than making an effort to practice speaking in Spanish. Building confidence in listening and writing often precedes confidence in speaking. The teachers can build on this success with listening comprehension and gradually increase the rigor and demand for oral communication in future lessons. Teachers can also use future research lessons to study and explore ways of minimizing use of English or written Spanish during oral language activities.

Preparation Matters

Perhaps the most distinguishing factor of this lesson was the careful thought teachers invested in designing scaffolded opportunities for listening and practice. Students identified the matchmaking activity, the rules for placement and use of the subjunctive, and the multiple opportunities for practice as instrumental levers that boosted understanding. Instead of doing three or four different activities just to keep students’ attention, each phase of the lesson prepared students for the next more difficult phase and advanced students toward a specific learning goal.

This kind of preparation and thoughtful progression of complexity can make or break a student’s confidence and willingness to actively engage in the acquisition process. The lesson reminds us that preparation matters. Students are the benefactors when they feel prepared to tackle challenging problems and tasks.






The Art and Habit of Inspiration

Dawn Hamby teaches 18 students in her Advanced Ceramics class. She designed the course to equip young artists with essential techniques for creating and critiquing 3-dimensional art. Using various tools and media (e.g., clay, plaster, wire, cardboard) students learn to exercise creative freedom and explore new possibilities for independent, original work.

Despite their extensive background and interest in art, Mrs. Hamby finds advanced students struggle with original thought, developing artistic identify, and self-generation of creative themes. Most students still revert to copying images and ideas they find online or making minor alterations to finished examples. Various scholars and publications address these challenges for the elementary level student, but limited resources exist for addressing the unique developmental challenges of the high school artist.  

Through this course, Mrs. Hamby hopes to foster student curiosity about important questions such as: How do artists work? What kind of skills and strategies do they employ to explore and express their ideas? How can I develop these skills and apply them to my artwork as well as other aspects of life? Most importantly, she hopes students will overcome their aversion to risk and cultivate new habits for originating and articulating ideas.

Practicing Inspiration

One aspect of learning how artists work is understanding the role of inspiration in art. In their own art projects, students may confine “inspiration” to finding and copying “cool” images on the internet rather than exploring their own artistic process and interests. Students may also presume that artists primarily find inspiration through dramatic moments of epiphany, while more often it’s the result of well-refined creative habits. Advanced artists develop a keen awareness of their own inspirational process and deliberately work through that process at the outset of a new project.

To help students move beyond such limited perceptions and cultivate more advanced habits, Mrs. Hamby planned a research lesson in second semester focused on artistic inspiration.

She began the class period with a warm-up writing activity and the introduction of three essential questions.

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She then asked students to think about the sources of inspiration for each of the recent class projects they completed. She emphasized the difference between copying other artists and “reverse-engineering.” She also provided a case study artwork example by Janet Echelman, titled “One Point Eight.” The title of this work refers to the length of time in microseconds that the earth’s day was shortened as a result of the 2011 earthquake and tsunami in Japan. Students studied the image as a class and took notes about five key process elements that contributed to the artist’s inspiration: subject matter, process, medium, context, concept.

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Click here to view these opening segments about sources of inspiration.

Mrs. Hamby continued the lesson with a sorting activity involving cards with artwork images from their own class projects as well as images from other artists. Students worked in small groups to discuss each card and arranged the artworks into categories based on inferences about the artists’ inspirations. Some students sorted cards by subject matter or visual style instead of thinking about the artist’s sources of inspiration, so Mrs. Hamby used probing questions about the artist’s intention. Each group shared their categories of inspiration with the whole class and recorded the full list of sources in their sketchbook notes.

Click here to view the sorting activity.

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Next, Mrs. Hamby asked students to explore personal sources of inspiration with a five-step brainstorming exercise resulting in a word-web of ideas for their next project:

  • Make a list of 20 things — What inspires you? What makes you want to make something? What makes you excited about coming to class? What do you wish you could do? What materials or techniques do you like to work with? What do you wish you could try? What do you love? What is a story you’d like to tell? If you could tell the whole world something, what would it be?

  • Circle Three ideas from your list above. How can you make these more specific? Write three ideas next to each one.

  • Word Web: From the three you circled, choose one inspiration that you will use for this project and place it in the center of the word web.

  • Branching out from main inspiration - List five five words that come to mind when you think of this of word.

  • Incorporating 3D design elements: How can you incorporate the elements of 3D design? Which principles will your sculpture focus on? How will your materials visually support your theme? How would you display this work?

Below is an example word-web from one student in the class.

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Finally, Ms. Hamby concluded the lesson by asking students to write a brief statement describing their next project. Examples included:

  • I want to make a sculpture of people that explores unity.

  • I want to make a mixed-media fish out of clay or foam exploring one’s mental health with beach trash as a metaphor.

  • I want to make a horse running that explores light and movement.

Click here to view the final lesson segments on personal inspiration.

Student Results

After completing the lesson, we analyzed students’ word webs and corresponding project statements based on three main criteria: creativity, heart, and courage. Below are the results of this analysis based on 10 representative samples.

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The results from the word webs reflected a wide range of diverse ideas and sources of inspiration. All but two samples showed clear evidence of proposed subject matter that was inspired by a topic of deep personal interest, and most of the final statements described projects with increased complexity, scope or scale.

Overall, the word webs revealed that students still need more assistance and practice moving beyond concrete topics of interest to elaborate on a concept or theme. But Mrs. Hamby was excited to see initial signs of students making connections between proposed subject matter and the emotion or message they hope to convey. She was also pleased to observe how the exercise helped students overcome “artist’s block,” and pushed them to unpack their own universe of ideas without copying examples from the internet. These findings were reinforced by student interview responses.

Student Interview

I: When you were grouping the cards into categories, what were some things you learned about where artists get their inspiration?

KC: I learned that you get inspiration from anywhere...Some of the categories we selected were nature, and shapes, and abstract objects, and people, like portraits of them, and also day-to-day life.

I: How has your approach to inspiration changed as a result of this lesson?

KC: I thought people thought of something randomly or something that impacted their lives. That was an eye-opener because lately I have been struggling with being inspired and what I want to make next...That was a lesson I needed right now. I’ve just been making whatever comes to me and that helped me dig deep and get inspired by different things.

I: How can we use an artist’s final product to understand the source of inspiration?

KC: First of all, what they made and what it’s made out of. Like the fishnet sculpture that Ms. Hamby showed us. I would have never thought that was made out of knotted fishnets. So the message that it’s sending and the different elements she used with that...like the movement of it...You can see how long it took...how they got there to make you feel a certain way about it.

I: How do artists approach the challenge of coming up with their next project?

KC: I think that like Janet and how she used the fishnet. I know that she was actually stuck and didn’t have her supplies so she had to totally come up with something new. Since I’m stuck with what I’m trying to make out of clay right now, I can maybe use different materials and be inspired to make something else like using wood or wire.

I: What have you learned about your own sources of inspiration?

KC: I feel really connected to nature...also I’ve really been into shapes lately and abstract things. And also colors and how they depict the emotion of it. Let’s say it’s all black and white or all reds or blues which make a tone of the piece.

I: Was there a moment or aspect of this lesson that helped you gain a new insight or understanding about the inspiration process?

KC: First we wrote down a whole bunch of stuff that we liked and I realized that I have a lot of choices and how we narrowed it down to one or two words. And how we went even more in depth choosing verbs and actions...Because I wouldn’t have thought about all the different things a horse could be doing. I was just thinking of making a standing horse but now I’m thinking of making it running, or eating...using light to show that it’s moving...I think that was helpful and I probably will use that in the future.

Reflections

Looking ahead, Mrs. Hamby hopes to build on the success of this lesson and continue working with students to envision a sculpture for their next project. Students will think about materials, techniques, form, and the principles of design. They will create sketches of the final product and make a plan for researching any new techniques or materials.

In addition, based on the results from interviews, observations, and work samples, at least three big ideas from the lesson represent compelling avenues for continued refinement and study over the long term:

  • Providing opportunities to learn from the inspiration journey of other artists--working backward through an artist’s story to study both the experiences and decisions that shaped the final product.

  • Reinforcing new academic language and for describing the process elements that contribute to inspiration: subject matter, process, medium, context, concept.

  • Expanding effective use of tools and processes for exploring and elaborating on their own world of ideas and interests (e.g., word webs and other methods).

These approaches helped equip students with a productive framework for exploring and understanding art--what it represents, what the artist was thinking, and how they might adopt similar habits to ignite and sustain their own creative potential.













Pictures Painted in Words: Discovering Imagery in American Literature

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Christina Perez has taught American Literature for six years. She continues to find great reward in helping students explore contemporary issues and culture through the lens of American history, themes, and stories. While advancing students writing and critical analysis skills, she believes the course also enables students to grow as social beings and view other’s life experiences with interest and empathy.

Over the last several years, one of the teaching and learning challenges Ms. Perez has faced is teaching students to identify and explain images in literature. “Unless it’s dripping in figurative language,” she explains, “they struggle to recognize, explain, and connect literary images to more important devices, and concepts for any given text.” Students are asked to use imagery throughout the course, but especially in second semester where students are expected to know, understand, and be able to readily navigate images when writing and articulating connections.

At the start of the 2018-19 school year, Ms. Perez tackled this challenge in her first unit with an emphasis on discovering imagery while exploring marginalized texts from Native American literature. She hoped to build a foundation in this first unit that would lead toward deeper understanding and analysis skills throughout the year, and ultimately prepare students for The Great Gatsby essay in second semester. We collaborated with Ms. Perez as she developed a research lesson for this unit.

Images and Empathy in Native American Lit

The first unit is called “Marginalized Literature.” The course focuses on the importance of all American literary writers, including Native Americans, specific to this unit. Native American authors have been historically absent from the literary canon and marginalized because of their spiritual and social distinctions, not to mention their physical differences. Students uncover the importance of what it means to be American in a society that is founded on immigration and racial/social disparity. As they read through various Native American texts, both ancient and current, these writers often demonstrate their identity amidst personal anecdotes and accounts of their lives as Americans, through imagery/image.

Some of the essential questions in this first unit include:

  • Due to mistreatment, how do natives view America?

  • What is the Native American dream?

  • What is the Native American lens and and their newfound identity?

Drawing Images to Discover and Connect

To scaffold students’ recognition and use of imagery while addressing these topics, Ms. Perez developed a new approach for students by introducing an imagery device chart (see example below), which requires students to think about the “pictures” that they see when they read.

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She asked students to draw, write, and articulate the images they found as they progressed through a given text. Specifically, she wanted students to be able to identify at least one image, depict the image on their charts based on the author’s description, and articulate its importance in relation to the purpose of the text. Students would then use this chart as an organizing device for their writing, drawing on class discussions and newfound knowledge to articulate connections in their explanation and analysis.

Lesson Storyline: Superman and Me

The storyline for this culminating lesson included the following major segments:

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1. Review examples of the “Native American dream,” covered in this unit. Discuss identities that have been assigned to Native Americans by society, and identities they aspire to.

Click here to view this opening review and discussion.

2. Read Sherman Alexie’s “Superman and Me” and ask students to annotate/make notes of important story elements.

Click here to browse through the narrative text.

Click here for a short video with student responses.

3. Work through example image as a class with basic drawing from teacher’s chart. Discuss purpose of the image. Discuss how the image connects to the Native American identity.

Click here to study teachers’ example chart.

4. Work in groups to complete these steps with another image from the text.

Click here to view a short example of a student group discussing the image of  “superman breaking down the door.”

5. Introduce example writing prompt for this text: Identify one image in Sherman Alexie’s “Superman and Me,” and explain its purpose (how) in defining the Native American identity. Be sure to tell me what is Alexie’s Native American identity.

Click here to view the teacher elaborating on this prompt during the lesson.

6. Study example response to this prompt and identify key components of an effective analysis.

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Click here to study the teachers’ example response.

7. Work in groups to generate an original response to the prompt using notes from the device chart.

Click here to watch a group of students composing a response.

8. Exchange responses with another group to offer peer review and suggestions. Discuss responses and image analysis as whole class.

Click here to view an example of students providing and receiving feedback from their peers.

9. Review and discuss purpose of the chart: Have you found this chart to be helpful? How does this chart prepare you for reading literature and explaining imagery? How might you use this process in the future?

Click here to view closing segment.

Student Interviews

Student work and lesson observations from classroom video showed that students made deep connections with the plight of the Native Americans and empathized with the challenges this marginalized population faced in forging new collective and individual identities as equally capable and intelligent members of society. Students also made great strides throughout the unit in learning to select, analyze, and articulate imagery from the literary texts. Excerpts from two students interviewed below provide rich examples of this deeper knowledge and understanding. Both students (Aaron Rivera and Avery Warren) were also featured in the video clips previously referenced. Aaron is the student in the black shirt that is composing a paragraph for his group directly in front of the camera. Avery is the student in the red shirt, sitting on the right, directly behind this first group.

I: What is an image in literature? How is it different from other ways we use images in everyday life?

AR: Images in literature are more abstract and you have to find the meaning yourself...in real life it’s more straight forward. An image in literature is a device used by the author to convey a message--sometimes it’s through symbolism.

AW: An image is written in words...The author has to find the descriptive words so the listener can get a feel of what’s going on...that it almost takes them to that place where they can see it.

I: What images stood out to you in this story? Which ones did you select to focus on as a group?

AR: I imagined the poverty of the reservation and I tried to picture the superman comic... In the comic, there is a door placed in front of him. He broke down the door and it shattered. He used that to tell a story as if he was superman breaking through adversity and barriers in his own life…

AW: We selected the image of superman knocking down the door. It’s a metaphor saying that they were trying to knock down their own door. They were given an identity and they didn’t agree with it, so he knocked down that door and created his own identity.

I: In your own words, what does it mean to be marginalized?

AR: We have been using that term in class to describe the adversity that Native Americans have to overcome. It’s when society can attach specific identities to you or a certain group of people. A lot of people, especially the settlers, saw Native Americans as unintelligent because they didn’t have technology and because there was a barrier in communication between them.

AW: Like a margin on the side of the paper, they were pushed to the outside. The Native Americans didn’t have a say in anything. They were outsiders that did not matter. I felt bad that this had to happen. It brought up not only literature but also history. It brought insights about how life was them back then...we see the other side of the story and how they lived in a time of oppression.

I: What did this lesson teach you about choosing an image and providing sufficient explanation/evidence to support your analysis? How has the chart helped you? Tell me more about that.

AR: In explaining and analyzing an image, the chart has helped me to structure my thoughts...it asks you to connect it back to the overarching theme of identify and that’s the part the chart has helped me the most with. I remember last year when I would write timed-writes it was hard for me to understand how to write an analysis properly, but with this chart it was more clear to me...I was only explaining what the image means but would not connect it back to the overall theme of my essay. The overarching theme of the story is sometimes not the same as the image you are describing.

AW: We’ve been using the graph/chart with the story title, the image, a picture of the image, the purpose, and the identify. It’s been extremely beneficial because it’s helped with organization. Analysis is difficult for me. You can look at that image and think back on it and go in more depth...consider all the factorI: When you exchanged feedback with the other group about your paragraphs, what insights did you gain about the analysis process?

AR: They told our group that we were choppy in the part leading up to the final sentence about how it relates back to his identity. We didn’t really tie all the separate information together. We just listed it and then expected the reader to do it themselves. We assumed the reader would know that the superman image represented the author as superman without ever pointing that out.

AW: We told the other group that they could have been more clear in connecting the explanation and purpose to the passage that they chose. Because if you don’t connect them it sounds like you are just putting all these sentences together. It doesn’t flow or connect.

I: Is there anything else from this lesson that you think is important that I haven’t asked you about?

AR: I think it’s important to know that the lesson has not just helped me with analysis in this format but it’s also helped a lot of other people I know. In writing the analysis, people struggled with organizing their thoughts and this chart helped people realize the order their thoughts should go in and how they should link them together.

AW: I think it’s important to realize the purpose of why we are reading these stories. It’s not only literature. This happened in the 1800s and it’s not something we should forget about. Everyone should be treated with equality.

Reflections

As Ms. Perez reflected on the lesson, she shared how pivotal it was to help students begin using the device chart not only to make a list of images in a story, but to employ it as a tool for structuring and preparing to compose a written analysis. In a subsequent assessment and writing assignment on “Burning River” (by Simon Ortiz) over 90% of the students utilized the device chart as a structure to plan and organize their composition. Some students showed substantial growth in analyzing and explaining images from the story with greater depth and examples from the text. Others showed marked improvement in connecting the images to the overall theme of identity. A few students showed advanced skill in both of these areas--unpacking the images and connecting them to the overall theme. About five students did not use the chart at all and left it blank while three others only used the chart partially. These students had weaker compositions with mostly summary and little original analysis. This provided evidence of the chart’s utility in scaffolding students’ analytical skills and writing. Ms. Perez plans to share this evidence with the class to help them reflect on their use of the tool.

In her own words, Ms. Perez offered the following insights and reflective comments for the lesson and described her plans for future instruction.

The process of analysis requires writers not only to read a work of literature and internalize the plot and author’s message, but it also requires the writer to think about the various elements that make up the piece. Specifically, in Sherman Alexie’s “Superman and Me,” my students were faced with a manifold of important images that address Native American identify. Students were not only expected to highlight important aspects of the story, but to identify images that illustrated the Native American dream. The chart helped each student organize their thoughts, focus on one image, and think about how that image demonstrated the Native American identity.

While it was evident students could think critically about each image identified, their ability to develop, explain, discuss, and connect their ideas to the bigger picture was lacking. This chart provided them with the opportunity to do just that. It provided a safe, yet organized space for each student to articulate their thought process without having to worry about verbiage and fluidity. Once the chart was completed, students could physically see their thoughts on paper. Once the writing began, they could take their articulated thoughts labeled in the chart and plug them in to their short responses. This would enable them to write the paragraph focusing solely on diction and flow. A task less daunting and much more organized.

Going forward, this chart will be invaluable for any writing format. As students continue in the next unit on Puritan Literature, the chart will prove helpful in organizing student ideas for The Scarlet Letter essay as they develop and discuss Hester Prynne’s character. While the original chart enables students to identify and think about the purpose of imagery, Christina plans to incorporate a new chart component for The Scarlet Letter focused on “character trait/quote of characterization,” followed by the “purpose” of why Hester Prynne’s character behaves in this manner, and how that “connects to the novel as a whole” (her place in the Puritan society). The outline of this chart will continue to help students organize their thoughts and identify important characteristics.

Christina looks forward to implementing this new component and assessing students’ continued progress as they build analysis skills and employ these tools on their own...even when not prompted by the teacher.





Pondering Parabolas

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Dustin Boburka teaches Enriched Algebra at Orange Lutheran. The course provides a critical foundation for Honors Geometry and Algebra 2 and helps students master key concepts and important reasoning skills essential for mathematics. The enriched curriculum also challenges students to develop problem solving skills for life and encourages students to explore the role of mathematics in the world around them.

Parabolas and Problem Solving

One example of these important problem solving skills is distinguishing and effectively using standard and vertex form for quadratic equations. Quadratic equations refer to equations with at least one squared variable. The graph of a quadratic equation always gives you a parabola.

The most standard form is ax² + bx + c = 0. The letter x represents an unknown, a b and c are the coefficients representing known numbers, and the letter a is not equal to zero.

The vertex form is represented by the equation: f(x) = a(x - h)2 + k, where (-h, k) is the vertex of the parabola.  

Both forms produce a graph with a parabola, but the starting point for solving the problems are different. Standard form provides the axis of symmetry but requires mathematical steps to find the vertex, whereas vertex form provides the vertex from the outset (h,k values). The following graphs from Desmos.com (which Mr. Boburka used with his students) provide a visual representation of how these equations correspond to graphs on a coordinate plane.

Standard Form: Example Graph

Vertex Form: Example Graph

While teaching these lesson on quadratic forms, Mr. Boburka specifically wants to help students address the following questions:

  • How are quadratic equations used in everyday life to calculate change and variation of quantities? What are some examples?

  • How are quadratic equations different from the linear equations you learned previously (e.g., the rate of change)?

  • How does the form of the equation (standard or vertex) influence your starting point for solving the problem?

  • For each type of quadratic form, what are the characteristics of the graph (parabola), and what steps are required to figure out the characteristics?

  • How do I plot those characteristics on the coordinate plane?

After teaching this course for twelve years, Mr. Boburka continually finds these concepts and skills challenging to teach and difficult for students to master.

Part I: What are parabolas and what is standard form?

Recently Mr. Boburka modified his approach for addressing these important topics by constructing a two-part lesson on standard form and vertex form. We collaborated with Mr. Boburka as he developed and implemented this research lesson.

He designed the first day as an opportunity to familiarize students with quadratic equations, how they are different from linear equations, and the kinds of graphs and parabolas that quadratic equations produce.

Mr. Boburka felt it was important to help students recognize that quadratic equations are not just formulas we memorize to solve mathematical puzzles. Instead he wants student to experience them as methods for solving problems used in everyday life, such as calculating areas, determining a product's financial profit, or finding the speed of an object. To that end, he started the lesson by sharing examples of life situations from Sciencing.com where change and variation of quantities are important. Examples included finding the area of a room, calculating a profit, throwing or hitting objects in the air for athletics, or estimating the speed of a kayak.

For the next several exercises in this first lesson, Mr. Boburka facilitated a class discussion by asking students to analyze example graphs and equations using the web-based graphing application Desmos. Since students can easily manipulate the graphs and equations in Desmos, they were able to better visualize and ponder the characteristics of various graphs and better understand the functions each graph represents.

He started by building on students prior knowledge and showing them example graphs of linear equations students had previously studied. He then continued with a guided-exploration of additional graphs and parabolas generated from quadratic equations in standard form. He specifically focused on helping students discover the axis of symmetry and teaching this as the pivotal step for equations in standard form. He asked the class to describe what they notice about the left and right side of the graph. One student commented, “They are the same.” Mr. Boburka continued eliciting comments by asking, “What does he mean by that--they are both the same.” Another student compared the left and right side to the identical wings on both sides of a butterfly.

Click on the link below to view a short sequence of clips from these opening segments and the guided-exploration of graphs and parabolas.

Day 1 Video Segments

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From the axis of symmetry, the class learned how to find the vertex and how to use the vertex to identify the y-intercept. They also learned how to interpret the ‘a’ value of the formula to determine which direction a parabola will open (upward or downard).

Mr. Boburka wrapped up this first lesson by giving students several more examples in standard form with Desmos, allowing them to first visualize each graph and then make explicit connections back to the equation.

Part II: What is vertex form? How is it different from standard form?

On day 2, Mr. Boburka began with a review of standard form and specifically asked students to recall the line of symmetry as the starting point in standard form. He contrasted this with vertex form which (consistent with its name) begins with the vertex of the parabola rather than the line symmetry.

He then facilitated several exercises in pairs, asking students to type various equations into Desmos and estimate what the vertex was for each equation.

Day 2 Opening Video Segment

He followed each set of practice equations with discussion where students identified characteristics of graphing in vertex form. Students discovered how different equations and different components (h value and k value) of the equations affect the placement of the parabolas on the coordinate plane. A change in the h value causes a horizontal shift and a change in the k value causes a vertical shift.  The class used x^2 and (x-2)^2 and (x+3)^2 to visualize a horizontal shift. And they used x^2 and x^2-2 and x^2+3 visualize a vertical shift.

Click on the link below and drag the slider for the h and k values to see how Desmos helped students visualize these characteristics.

Desmos - Visualizing h and k values

Building on these visual insights and connections between the equations and graphs, Mr. Boburka now increased the rigor of the task by asking students to study three new equations. Specifically, he instructed students to analyze the h and k values and identify the vertex before graphing them in Desmos. This time they used Desmos to check and confirm their answers rather than using it to identify the answers.

The next step of scaffolding was asking students to explain how they ascertained the vertex in these equations without using Desmos. The key stumbling block here was helping students see that the negative next to the h value in the equation causes a horizontal shift in the opposite direction of our intuition, while the positive next to the k value causes a vertical shift that is more intuitive.

f(x) = a(x - h)2 + k                  

Click below to view a clip where a student articulates this key point.

Day 2 - Student Insight

Following these introductory exercises for vertex form, the class spent the rest of the period working to identify other characteristics of the graphs including the y-intercept, maximum, minimum, range, domain, width, and mirror points.  

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Mr. Boburka concluded class by emphasizing the value of Desmos as a tool for confirming answers and obtaining feedback during individual homework and practice.

Click on the link below to view a few clips from these last segments of Day 2.

Day 2 Closing Segments

Student Interviews

The excerpts from two students interviewed below provide helpful evidence of the student introspection and insight fostered by these learning opportunities. Both students, Logan Mills and Christa Barksdale were also featured in the video clips. Logan shared the analogy of the butterfly for the axis of symmetry and Christa is the student next to him in the video clips who answered a question about the vertex in one of the sample problems.

I: Describe some of your thoughts at the beginning of this lesson? What was your initial impression of quadratic equations?

C: At first I was kind of confused. I was thinking about how they incorporate into our everyday lives and I didn’t understand how all of that worked. But when we used the app Desmos that helped show me. Seeing it on the screen and putting in the equation and calculating it and seeing how it worked helped me understand a little bit better. The golf ball made sense to me...that put a picture in my head...there is one [specific] point where it starts to come down.

L: I was a little bit overwhelmed...that we have to learn a bunch of new formulas...how to solve for the parabola and how to find the vertex...I remember he said they are used for business like charting profit...I like how he was able to relate it to real world jobs.

I: How does the form of the equation (standard or vertex) influence your starting point for solving the problem?

C: For standard form, you look to find the a, b, & c and you have to figure out the vertex. For vertex form, you just find the vertex in the parenthesis with the x. The vertex form seems much easier to me.

L: You build the graph a different way. One thing that really helped me tell them apart was the vertex form had the parenthesis...You go straight to find the coordinates, and then find the vertex.

I: Tell me your impression of the Desmos app. What did you find helpful about that?

C: I found it really helpful to put in the exact equation that was given to you and it shows you what it looks like...I could pinpoint where the line intersects with each point. I think it’s a good app to check...it gives you confidence if you did get the right answer.

L: I love Desmos because when you are done graphing you can put the original problem into Desmos and see what the perfect graph should look like...If you got it wrong you can go back and fix it. When I look at Desmos I can see where the line [of symmetry] is and I can make sure I find the exact coordinates...I also see where the min and max point are...and if it’s in the wrong spot I know I have a problem.

I: Do you recall any particular moments in the lesson (either Day 1 or Day 2) where you felt like you reached a new level of insight or understanding about these concepts?

C: I remember when learning the vertex form, how the h is always opposite and the c, k, you just put them down. I also remember one day I was working with Logan. We were working with standard form. We were trying to find the y-intercept and we didn’t square it. I had to sit back and look at it before I could understand.

L: The butterfly...it just popped into my head...because each wing is identical if you split it down the middle. What I had to do was fine the [line of symmetry] and the min and max point on the parabola had to line up. It wouldn’t have the butterfly effect...if it wasn’t on the [line].

I: Take a moment to read these examples again from the beginning of the lesson. What insights do you have now about how quadratic equations are used in everyday life?

C: Reading about the sports again...like throwing the ball to your friend, how far away they are and how high you throw it so it comes down...you have to know the height so it will come down like a parabola or curve. And also like building...for an area of the room. You have to know the length, width, height and how big it is. I always thought that was just geometry...that isn’t algebra...but it talked about figuring out how big the wood is to make sure it will fit. If you only have four square feet of wood, you have to do 2x squared and if it’s less than or equal to 4 you could use it, but if was greater, then it wouldn’t fit.

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L: The first picture has the throwing of the javelin and I think that’s an even better analogy because a javelin is like a long stick that you have to throw up...and you have to calculate the angle...If you throw it straight up, it will come right back down. Knowing that--it does help, because I’m a triple jumper...and so I have to take off at three different jumps and end up in the pit. The first jump I take has to be a little bit more vertical than the second and I have to take off at about a 45 degree angle...the second one I go directly forward so I get more momentum and the third one is like a very big parabola...I have to go really high to get a better a distance.

Reflections

As Mr. Boburka reflected on these lessons and the student results, he shared his own observations and insights as well as plans for future instruction. During pair work and class discussions he was pleased to find evidence of students correctly using terminology to describe graphing characteristics and solutions. Understanding and utilizing key mathematics vocabulary is an important first step in grasping concepts of graphs and parabolas.

During both lessons (standard form and vertex form), he also observed how the large number of characteristics associated with each quadratic form presented a significant cognitive load for students. While identifying and describing each characteristic is important, he felt it would be beneficial, especially at the Algebra II level, to feature more prominently the vertex and axis of symmetry as key components for analysis regardless of which form students are using.

Another insight he identified related to ways he might be more intentional about engaging students with central ideas in mathematics. Going forward, Mr. Boburka hopes to find key points in each chapter where he might intentionally facilitate opportunities for students to explore, think, and contribute to classroom discussion about the use of math concepts in everyday life. As opposed to just sharing examples, he wants to help students generate and analyze their own prior knowledge, observations, and ideas.

Finally, Mr. Boburka reflected on the pivotal moment in this lesson when Logan brought up the analogy of the butterfly for the axis of symmetry. He expressed the importance of being patient and providing time for students to contribute, struggle, and share their thinking. Too often we rush rush to cover content while students rush to solve problems. Recognizing how significant this was for both Logan and the class, Mr. Boburka plans to be more intentional about creating similar opportunities for open-ended dialogue and discussion.





You're free! What's your next move? A HyFlex journey through the Civil War Reconstruction

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Peter Lark has taught US history for 19 years, long enough to witness changes and additions to the latest chapter of our nation’s story. Throughout his career, Mr. Lark has consistently grappled with the challenge of covering this full breadth of expanding content while also constructing rich and memorable learning opportunities where students can discover, apply and transfer ideas about history to their everyday lives.

Mr. Lark views US History as a critical course for learning not only the storyline and content of our nation’s journey but also for helping students understand how actions and events of the past influence beliefs and issues we face today. It also helps them learn that people will hold diverse opinions about life and politics. These opinions, while different from their own, may be supported by equally good reason and rationale.

This year, Mr. Lark is teaching his first HyFlex US history class. A HyFlex course blends face-to-face (f2f) learning with a flexible learning session (FLS) that is technology enhanced and primarily self-directed. It allows students to set the pace for a portion of their learning while still providing opportunities for face-to-face collaboration and guaranteeing access to individual or small group assistance from their teacher within the school day. Teachers can also use flex sessions to reduce class size and optimize facilitation of group work by bringing in one group or a smaller set of groups. Mr. Lark has been working to leverage the unique design of HyFlex to foster what he describes as a “US History Lab.” One insightful example is a research lesson he taught on the Civil War Reconstruction. We collaborated with Mr. Lark as he developed and implemented this lesson.

More than a Series of Battles

When students first think of the Civil War they often think of generals and soldiers, guns and military strategy. They think of images so often captured in popular films or TV series about the North and South with men dressed in blue and grey uniforms, officers on horses, cannons and bayonets. Mr. Lark’s goal in teaching the Civil War, and particularly the reconstruction period following the war is to help students look beyond the military battles and victories and grapple with the deeper issues framing the period. Key questions Mr. Lark wants student to ponder include

  • What caused this war between North and South?

  • What was resolved and accomplished at the war’s conclusion?

  • What complexities did the nation face in bringing the South back into the Union?

  • What does this teach us about our government, society, and culture?

  • How do the outcomes of this war still affect our contemporary lives and government?

  • How do the failures of this reconstruction period linger on in the lives of African Americans today?

Part I: The US History Lab

To aid comprehension of these profound historical themes, Mr. Lark constructed a two-part lesson that guided students through the emotional journey and complex challenges of this critical period. Using the unique structure of his HyFlex class, he designed the first day as a US History Lab where students worked in teams of three to four and traveled through a series of stations focused on the reconstruction era. He divided the flex session into half (approximately 35 minutes each) and assigned a few teams to each half of the period. Click here to view a one minute introduction to the flex session and the station exercise Mr. Lark designed.

Students then spent seven to eight minutes in each of the stations described below.

Station 1: Setting the Stage: Who was Roger Taney? (7-8 minutes)

Station format: Students review a news clipping from the recent statue removal of

Roger Taney and provide a written description of Taney's decision about Dred Scott.  

Reflection Questions:  Who was Roger Taney? What did he decide in the famous Dred Scott Decision (1858)? When did the Civil War begin?

Station 2: The Civil War: What was won? (7-8 min)

Station format: Students watch a short documentary video describing the historical significance of the Emancipation Proclamation.

Reflection Questions: What was won as an outcome of the Civil War? Watch the clip, review the images and think as a group. With your group, think about and rank the most important outcomes of the war? Why did you rank each this way?

Station 3: What did war leave behind? (7-8 min)

Station format: Students review assorted images: Newspaper headlines on Lincoln’s death, pictures of people with amputated limbs, burned path through Georgia, and bent rails on train tracks.

Reflection Questions: What was left behind by the war? Use the images to aid your thinking about what the war left behind. What was left behind in the North? What was left behind in the South?

Station 4: You’re free! What’s your next move? (7-8 min)

Station format:  Students study contrasting images of a freed slave with a jubilant expression juxtaposed to the image of freed slave with a despondent expression.

Reflection Questions: The war is over and you are now a freed-slave. What would you choose to do next? How would you go about doing that?

Click below to watch two example student groups contemplating the images and reflection questions at Station 4 (You’re free! What’s your next move?”).

Group #1:  “I think it would be scary.”

Group #2:  “Slavery under a new name.”  

Unlike his traditional class, the flex session enabled Mr. Lark to focus his attention on just three groups of students at a time (half of the class) and doubled the amount of energy and attention he could invest with each group as they progressed through the stations. In the clip from Group #1, for example, Mr. Lark was able to monitor the distribution of talk among team members and strategically draw out important insights from one girl in the group who had been quietly listening but not contributing much to the discussion. Similarly, in the clip from Group #2, he was able to listen-in during a pivotal moment, reinforce the group’s emerging insights about the South “reinventing slavery,” and build some anticipation for key terms and ideas they would discuss in the subsequent class period.

One of the goals of small group work is to help “make students’ thinking visible” so the teacher can better understand, probe, and nudge forward student thinking as they wrestle aloud with important questions. Mr. Lark believes these types of exchanges are critical opportunities for helping students make memorable connections with the content. Combining the HyFlex format with the station design increased the probability of those exchanges and removed the pressure Mr. Lark normally faces in managing the entire class while also circulating to facilitate deeper thinking. The result was deeper reflective discussions among groups at each station which  prepared students for deeper analysis and study in the subsequent class period.

Click here to watch Mr. Lark wrap up the station work and build a bridge of anticipation for the upcoming f2f lesson.

Part II: Bringing it All Together

During the next regular class period following this flex lesson, Mr. Lark organized the classroom into groups based on notes they recorded in their station work. He strategically distributed students with classmates that were on different teams during the previous lesson so they might gain new insights and perspectives. He reframed and reintroduced the key questions from each station and facilitated a whole group discussion interspersed with opportunities for small group sharing and exchanges. Before and after reintroducing each question he elaborated on key events and themes from the period to deepen their understanding and insight and to aid students in making connections to the present day.

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Click here to watch a video clip from Day 2 where Mr. Lark guides the class through a deeper analysis of the questions from Station 1:  Roger Taney and the Dred Scott decision.

Using this same approach, Mr. Lark continued working through the key themes and ideas for each of the four stations. He wrapped up the exercise by asking students to ponder one additional question. “If you were an African American living at this time, what would you hope for?” Students talked about equality, mobility to get out of the South, more diversity within communities, and more help from the government. Mr. Lark pointed out that the government, up until this point, had applied a very strict interpretation of the constitution and had not played an active role as a change agent in society. The Civil War changed that, he explained, resulting in new funding and initiatives such as the transcontinental railroad, the homestead act which opened up the Western Territory for settlement, and the establishment of many state universities.

Finally, Mr. Lark transitioned from this discussion to a more detailed explanation of the key terms and historical milestones of the era, elaborating on the political, social, and economic factors Americans faced as they struggled to reunify the nation.

Student Interviews

Observations of students during station work and throughout the f2f lesson on Day 2 revealed a significant level of reflection, introspection, and empathy for the challenges Americans, and specifically African Americans, faced during this time period. The excerpts from two students interviewed below provide additional evidence of that introspection and insight. Both students (Avery Seagren and Kyle Hill) were members of Group 2 featured in the previous video clip.

I: As you traveled through the stations during that first lesson, what were some of your thoughts, feelings, and reactions?

A: I thought it was interesting the way we went about it. I like that we got to hear other people’s opinions...I thought it was nice to have the small groups...just to condense it...you feel like there is more discussion going on between you and the teacher.

K: I found the Roger Taney article very interesting...I saw how a guy was just following the trends of that time and technically making the right decisions became a villain for that one thing he did. He was not at the forefront of racism...but because of that one trial, he became this figure that people had painted to be a huge racist.

I: As you worked through the station about “What was won?” what were some of your impressions?

A: I think there were split opinions...back then...I think it was fantastic...especially the 13th Amendment. It was interesting to think about the Emancipation Proclamation. I didn’t know it was only the rebelling states’ slaves that were freed…

K: Out of all the war and violence what was born was the hope that all types of people might be treated equally without the fear of being targeted by bigotry and racism…the end of the Civil War is a start of a new beginning...what was won was hope that everyone might dwell together in unity.

I: What else did you realize about the time period and circumstances as you also reflected on “What was lost?” And “You’re free. What’s your next move?”

A: Definitely there was a lot more lost in the South than the North. It was interesting to learn that one of the Northern tactics was burning everything in their wake when they were sweeping through the South...it was sad to see all the destruction, because that was their economy.

K: I’m sure most of the slaves were excited to be free, but I’m sure many were also feeling pessimistic about leaving the plantation where they had shelter, food, and clothing. Now they are thrown into a world that doesn’t see them on the level of the rest of society. So what are they going to do? How are they going to survive? I’m sure some of them still felt enslaved...I think fear was a huge thing.

I: How does this lesson about the Civil War Reconstruction relate to the way you think about life in America today?

A: I think all of the violence over racism today...I feel like we should have learned from history. I definitely reflects the same pattern.

K: During the [second] class period, Mr. Lark talked about how the churches were segregated. I think it’s interesting how that has translated over time...kept that same thing with one race being predominant in a church. I think we have that time period to blame. There is still a separation of races.  

I: One of the things you learned in these lessons was that the Civil War Reconstruction Era was a time period that was focused on change. What can we learn from this time period about how

change happens in society?

A: Mr. Lark had said that change sometimes takes people dying, a generation that has a certain opinion to go away...I definitely agree with that. People just are so firm in their belief that they aren’t open to another opinion...those people are the ones blocking progress.

K: Well...it takes forever. I think that’s something Mr. Lark was also making clear...that change doesn’t happen in a few years. It takes...decades.

Reflections

As Mr. Lark reflected on the lesson, he shared his own observations of students’ journey through the lesson content. He was gratified to see how students connected with the emotions of the period, how they stepped out of their own world view to consider the complexities from multiple perspectives. He was excited to see their level of reflection on the questions he posed as they made connections between the challenges of the Reconstruction era and the pressing issues in society today. A week later, after looking at their essay responses on a Civil War test, he was thrilled to see the highest class average he has ever experienced in this course.

Mr. Lark also reflected on his new “Lab” experiment with the HyFlex model and the opportunity this afforded for more focused interaction with a smaller number of students. He is looking forward to adapting that design for future units throughout the year as he continues to balance presentation of key facts and content with rich and memorable learning opportunities for discovery and application.




Let’s Have S’more Chemistry: Marshmallows, Chocolate, Grams, and Moles

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Tanya Grasz teaches Honors Chemistry. Students who enroll for this class have demonstrated solid mastery of biology and algebra as well as a capacity for problem solving and critical thinking. The subject matter is demanding. Even well-prepared students often struggle with the advanced content as concepts build and converge over time with increasingly complex applications.

Ms. Grasz approaches this challenge with the same spirit of inquiry she expects from her students as they engage in ongoing investigation of scientific phenomenon. She is energized when students grasp an important lesson or idea but also perplexed when they struggle to master key concepts after careful teaching and assistance. One example is her ongoing efforts to help students use mole ratios when solving stoichiometry problems.

Brief Intro to Stoichiometry

Equations are a chemist’s recipe. Equations tell chemists what amounts of reactants to mix and what amounts of products to expect. When you know the quantity of one substance in a reaction, you can calculate the quantity of any other substance consumed or created in the reaction. For example, if we have the reaction N2 + 3H2 → 2NH3,  we know that nitrogen and hydrogen molecules will react in a 1:3 proportion. The calculation of quantities in chemical reactions is called stoichiometry and mole ratios are the bridge for converting between various units of chemical quantities.

Teaching and Learning Challenge

Ms. Grasz found that students typically struggle with several key aspects of solving stoichiometry problems as well as general principles of dimensional analysis:

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  • Interpreting and setting up problems

  • Understanding where the problem resides on the “mole map” (realizing what they are given and understanding where they need to end up)

  • Understanding and using mole ratios as the critical bridge for converting units of one substance to units of another.

  • And, most importantly...moving from a formulaic pattern (calculations of units) to a conceptual understanding of mole ratios.

Bridging the Gap

Ms. Grasz teamed up with another science colleague, Jill Ronstadt, to investigate instructional solutions. They planned and implemented a research lesson with the following components.

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First, they used a cooking analogy to help students grasp the proportional conversion process with something familiar and accessible from everyday life. “A recipe is like a balanced chemical equation,” Ms. Grasz explained. “Ingredients are the reactants. Cookies or cake or burritos are the products.”


She then asked students to think through an example recipe for S’mores. The goal was for the students to understand that there is no way to know how many S’mores could be made given a certain number of ingredients if there is no recipe to follow or refer to.

Video Clip #1 - Chemistry is Like Cooking

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Moving from food to chemistry, Ms. Grasz helped students review what the coefficients in the chemical equations represent using a relatively simple equation for ammonia (NH3) production. They discussed why the coefficients do not represent grams, but instead represent moles and molecules or sometimes liters (for problems using gases at standard temperature and pressure). This is why using mole ratios are so important--they bridge the gap between various units in the conversion process.

Next, Ms. Grasz asked students to work on a stoichiometry problem. The problem asked students to move from grams to moles. “How many grams of ammonia are produced from 0.60 moles N2? “ She gave students time to work individually. While they were working, she took inventory of students that solved the problem successfully and those that made a wrong turn somewhere in the process. She found that 8 of 27 students had solved the problem correctly. She then divided the class into eight groups and asked students who were successful to teach and discuss with others how they solved the problem. Ms. Grasz wrapped up the lesson by reminding students that the equation is their “recipe” and the numbers for the mole ratio are coming from “the coefficients of a balanced chemical equation.”  She then assigned additional problems for homework, including several questions about a recipe for chocolate chip cookies, and two problems involving chemical equations.

Video Clip #2 - Practice with Stoichiometry

Day 2 Results

The results from homework showed that nearly all students were able to solve problems about chocolate chip cookies, but most still had questions about problems using chemical equations. “They knew the mantra about using their equation to find the mole ratios,” Ms. Grasz explained, “but had not connected all the dots yet.”

Ms. Grasz spent the next class period persisting to help all students “connect the dots” and learn to use the balanced chemical equation as their “recipe” for solving stoichiometry problems. She guided the class through the homework examples. She paused at each phase of the problems to elicit student thinking and asked them to direct the solution process. She then gave students a new problem to solve independently and circulated around the room to document student progress. While circulating, she strategically placed a dot on individual papers to keep track of where students were struggling or asking questions. She then collected the papers to better diagnose the primary areas of difficulty. During this second round of work, Ms. Grasz’s records showed that of 27 students in the class, all but 8 solved the problem successfully and 5 of these 8 failed to correctly use the mole ratio. This was a substantial improvement from Day 1 where only eight students solved the Ammonia problem correctly. Ms. Grasz continued with several more problems at the end of the period until the entire class could use the “recipes” effectively.

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The excerpts below from student interviews describe this journey of discovery and deeper understanding from two students’ point of view. Both students were members of the same small group featured in the second video clip above. In the video, Cove was the “teacher” for the group. Brandon was the student on the far right, opposite side of the table.

Interview with honors chemistry student, Brandon Washiashi

I: When you were first asked to work on that ammonia problem, what process did you use to solve the problem. Where did you start?

B: We were coming off the previous test which covered moles and if I remember correctly, I think I tried taking ammonia and turning into moles…

I: How did that work for you?

B: Well, I got the wrong answer.

I: What were you were struggling with?

B: The numbers...I didn’t know the ratio and it’s kind of hard to calculate when you don’t know how much to calculate.

I: What did you understand about the s’mores analogy at that point?

B: I didn’t think to look there...It wasn’t on my radar until after the revelation of the ammonia problem.

I: What did you learn from the conversation with Cove and the group work?

B: That I need the correct amount of coefficients to make the correct number of ammonia.

I: After that, you had some homework problems. How were those?

B: The first one I was still getting into...I still had a couple of questions...after that I’ve been fine.

I: Was there anything that Cove said, that Mrs. Grasz said...where did it start to “click” for you?

B: I used the map...that was a big help...making sure it’s a balanced equation, of course, plug it in, and follow the bridge. For me, it was pretty straight-forward after that.

I: Where do you think your biggest gap was prior to that?

B: All goes back to the ratio...I can can’t cross unless I know the ratio.

I: How is this class affecting the way you think about science?

B: Well, I’m definitely thinking about it more mathematically now...Knowing all of the previous stuff is helping, because if I didn’t know how to look at a periodic table, I wouldn't be able to find the atomic mass of Oxygen. If I didn’t know Oxygen was a diatomic molecule I would just look at it as...one atom instead of two. If I didn’t know about moles, I couldn’t convert into moles and cross the bridge and figure out the problem.

Interview with honors chemistry student, Cove Carlson

I:  What were you thinking as you listened to Ms. Grasz explain the s’mores analogy?

C: We just finished a unit on conversion factors...I was comparing it to [moles]...I was piecing together as well, the balanced equation--the coefficients played a part...It was easy to picture the ratio. I make s’mores...so I can picture how each one corresponds to the other.

I: When you were first asked to work on that ammonia problem, tell me about your thinking.

C: I’m not a very good math person, so I know I was a little nervous. I wasn’t expecting to get it right, but I did have a sensation that “this makes sense”….I was looking at what I was given and thinking back to what she taught us with s’mores...There’s 0.5 chocolate bars to two graham crackers...there’s two ammonia to whatever it was in the equation…

I: Was there anything in particular that Ms. Grasz said that helped you?

C: It really stuck with me that the coefficients are an important part of the balanced chemical equation.

I: During the group work segment, how did you feel about teaching the group?

C: ...I felt a little proud...I got this. It makes sense to me.

I: Can you remember more about what was making sense to you?

C: ...Before it had always been one mole. This equals one mole; one mole equals this. The biggest thing I understood was that now it wasn’t one mole. It was two, three, four, however many--the coefficients of the chemical equation.

I: How is this class affecting the way you think about science?

C: Before going into this class, I was thinking, “I’m not a science person...” After this, I was grounded. This is a solid base for me. I know how to do this. I can keep building off this...You got to be persistent...You might spend three or four hours on an assignment, but if that helps you get it, everything else becomes so easy, because it all builds off each other.

Reflections

As illustrated in the student comments above, successful classroom teaching is rarely a silver bullet or single magical strategy. More often, it’s the hard work of carefully stringing together a sequence of subtle, incremental innovations that help students advance toward deeper understanding. Ms. Grasz’s stoichiometry work is a great example. She helped students access prior knowledge and make connections to the work done in previous chapters. She developed a compelling analogy with the s’mores recipe. She strategically grouped students so they could learn from each other’s problem-solving efforts. She gave them multiple opportunities for practice and feedback. She diagnosed where they were struggling and scaffolded their learning through guided questions and targeted assistance. She helped students remember to start with their recipe--the balanced chemical equation--and she helped them learn to use the “mole ratio” as the bridge between reactants and products.  

Ms. Grasz’s persistence with teaching fueled a parallel persistence with learning, enabling students to grapple with difficult problems and master core concepts of stoichiometry. Ms. Grasz and Ms. Ronstadt are now mapping out their next project to study and improve teaching and learning. Stay tuned for s’more examples and findings from teaching and learning at OLu.