(This is the second post in a four-part series. You can see Part One here.)
The new question-of-the-week is:
What is an instructional strategy and/or teaching concept that you think is under-used/under-appreciated in the classroom that you think should be practiced more widely?
In Part One, Kathy Glass, Amber Chandler, Carol Salva, Jennifer Davis Bowman, and Janet Allen proposed their “nominees.” You can listen to a 10-minute conversation I had with Kathy, Amber, Carol, and Jennifer on my BAM! Radio Show. You can also find a list of, and links to, previous shows here.
Today’s contributors are Jo Boaler, Katie Brown, Rachael George, Laura Greenstein, Dan Rothstein, David Jacob, and Greg Brown.
Response From Jo Boaler
Jo Boaler is professor of mathematics education at Stanford University, co-founder of youcubed, and author of Mathematical Mindsets: Unleashing Students Potential through Creative Math, Inspiring Messages and Innovative Teaching. Contact: email@example.com:
For centuries, mathematics classes have focused on numbers, expressions and equations. This is especially true of elementary school mathematics and the algebra taught in middle and high school. But new brain research is showing that our brains think visually about mathematics and even when we perform a bare number calculation five different pathways are involved, two of which are visual (Menon, 2014). When students are asked to visualize while studying mathematics, their achievement and engagement increase significantly (see for example Reimer & Moyer-Packenham, 2005; Boaler, 2016).
Asking students to think visually and encouraging them to come up with their own visual representations is different from seeing a visual representation given by a textbook or teacher.
In any mathematics teaching situation, a teacher can ask students how they see the questions and ideas, and prompt them to represent their thinking visually. Teachers could, for example, ask students to come up with different ways to solve 18 x 5, and then show them examples of visual representations of the equation, such as those below. Teachers can then ask students to come up with their own visual and numerical solutions to other questions.
This example is also shown in an animated video for use with students.
A highly effective teaching strategy is the use of “dot card number talks,” in which teachers show a collection of dots and ask students how many they see and how they have grouped the dots to see them. This not only illustrates the visual nature of mathematics it values the multiple different ways that students see and experience mathematics. An example of me teaching a dot card number talk to sixth grade students can be seen here.
Additionally, working with a number-line representation of quantity has been shown in cognitive studies to be extremely important for the development of numerical knowledge and a precursor of children’s academic success (Kucian et al., 2011; Hubbard et al., 2005, Siegler & Booth, 2004; Schneider et al., 2009).
Researchers found that after four, 15-minute sessions of playing a game with a number line, differences in knowledge between students from low-income backgrounds and those from middle-income backgrounds were eliminated (Siegler & Ramani, 2008).
Separately, new research explained more fully in Boaler & Chen (2016) also shows the critical importance of using fingers to improve mathematical understanding.
Algebra classes are often dedicated to students rearranging symbols. Students approach important mathematical concepts, such as functions, through numbers and equations, without any visual understanding. In my own teaching (which can be seen in this three-minute film), I approach algebra visually, as well as numerically and symbolically. In one activity, for example, inspired by Cathy Humphreys, I ask students to look briefly at a border around a square and work out how many squares are in the border, without counting them (see also, Boaler & Humphreys, 2005):
The students think about the number of squares in the border in many different ways, shown below. I ask them to think visually, then numerically and eventually algebraically.
To engage students in productive visual thinking, you can ask them at regular intervals how they see mathematical ideas and invite students to draw what they see. As students think about visuals and numbers together, they will exercise different brain areas and encourage communication between brain areas. Growing evidence suggests that communication between brain areas is critical for brain growth. Researchers recently concluded that what distinguishes ‘geniuses’ from others is the additional communication they have between different brain areas (National Geographic, 2017).
When mathematics classrooms focus on numbers, status differences between students often emerge, to the detriment of classroom culture and learning. Some students declare that the work is “easy” or “hard,” or announce they have “finished” after racing through a worksheet. But when the same content is taught visually, it is my experience that the status differences disappear. It seems possible that visual mathematics may contribute to equity, in valuing students’ thinking in different ways, as well as encouraging deep engagement. I have found that all students I teach are excited to see mathematical ideas, and from there they develop higher levels of understanding and performance (see also Boaler, 2016).
A common misconception in education is that drawing, visualizing or working with models and manipulatives is low level or only for young children. Yet some of the most interesting and high-level mathematics is predominantly visual. Maryam Mirzakhani was the first woman to win the coveted Field’s medal, the greatest prize in mathematics. Her work was almost entirely visual. One of the tasks we share on our youcubed.org site, is dedicated to the memory of Maryam, who died recently, and introduces her mathematical ideas to a K-12 audience:
Some of Maryam’s drawings from Quanta magazine video.
Other mathematicians described Maryam’s important theories as “beautiful,” “stunning,” and connecting previously unconnected theories in mathematics. Many children go through hundreds of hours of calculating, only ever seeing numbers and symbols. But mathematicians rarely, if ever, solve a problem without visual representations. As West reflects: “It’s masochism for a mathematician to do without pictures” (2004, p 27).
Until recently workplace knowledge was based on words and numbers. Today, knowledge is based largely on images, which are ‘rich in content and information’ (West, 2004). Most companies now have large amounts of data—typically referred to as “big data"—and many of the fastest-growing jobs involve making sense of the data and representing data patterns visually. Computer scientists and mathematicians at Stanford University and elsewhere now see patterns in data that could never have been picked up by statistical techniques.
I recently spent time at a local middle school near Stanford, where we trialed a set of visual mathematics activities that we had developed and shared on youcubed.org. At the end of the week, a parent stopped me and asked what we had done in class. She told me that her daughter had always told her she disliked math and couldn’t do it. But after working on our visual tasks, she came home and said she had changed her mind and could see a future in mathematics.
Why? The mathematics was open, creative and visual.
Every year we give away a free “week of inspirational math” on youcubed.org—a range of open, creative lessons for K-12 teachers and students. The 2017 lessons can be downloaded here. In previous years the lessons were downloaded over 3 million times and praised by students and teachers.
When teachers encourage and celebrate the many ways students see and think about ideas, mathematics classrooms become more exciting places—for teachers and for students. Visual mathematics activities promote deep engagement, new understandings and visual brain activity. Perhaps most importantly, they also show students that mathematics can be an open, creative and beautiful subject that they can appreciate, see and understand deeply.
Note: For more details on the visual nature of mathematical understanding readers may go to //www.youcubed.org/resource/visual-mathematics/ or read Boaler, Jo (2016). Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching. Chappaqua, NY: Jossey-Bass/Wiley.
Editor’s Note: The reference list for Jo’s response was a lengthy one. Instead of publishing theme here, or you can find them on this downloadable document.
Response From Katie Brown
Katie Brown is an ELL Specialist in Bellingham, Wash., and the 2014 Washington State Teacher of the Year. She works with students, teachers and families to best meet the needs of language learners:
We know that the teacher-student relationship is imperative for learning. Students need to feel safe and competent academically, emotionally and socially. As Rita Pierson reminds us in her famous TED Talk, Every Kid Needs a Champion: “Students don’t learn from teachers they don’t like.” There is nothing more important in education than building rapport and trust with our students.
One teaching strategy guaranteed to build positive rapport is greeting students at the door. Every day.
Greeting students at the door may seem simple, but how many teachers actually do it? The realities of preparing lessons, organizing the classroom, checking e-mail, problem solving tech glitches, making copies, and conferring with students usually suck up all the time before class. These tasks often take priority over standing in the doorway before that first student arrives. But, greeting students at the door is well worth the effort.
Greeting students at the door sends many important messages. First, it starts the day off with a positive interaction. Greeting students with a smile and warm welcome shows them that today is a new day, regardless of what may have occurred yesterday. Second, this simple act allows teachers to connect with each student personally. You can quickly gauge the emotional state of your students and have a one-on-one chat about their lives. This positive interaction supports the emotional needs of our students so they can engage in learning.
Everything we say and do (or don’t say or do) communicates a message. Think about a time when you entered a room for a meeting or an event and you were not greeted by anyone. It feels uncomfortable and isolating. Now think of a time when you entered a room and were greeted immediately. When people are excited to see you, you feel welcome, your feelings of uneasiness decrease, and you feel more motivated to engage with others.
By now, you have most likely watched the viral video of teacher Barry White creating personal handshakes with each of his students. Now that’s an inspiring personal greeting at the door. We don’t all have to be this creative (nor should we), but the rationale remains the same. Greeting students at the door is another way to say, “I care about you,” “You come first,” “I’m excited you are here”, “I see you,” “You are welcome here,” “I trust you,” “We are a community,” “Let’s have a great day.”
Greeting students at the door is an ideal strategy to bond with students regardless of their age or grade level. It starts the day with feelings of inclusion, safety, trust and joy. And we all could use more joy in our lives.
Response From Rachael George
Rachael George is a member of the ASCD Emerging Leaders Class of 2015 and currently serves as the principal of Sandy Grade School in the Oregon Trail School District. Prior to serving as an elementary principal, George was a middle school principal of an “outstanding” and two-time “Level 5: Model School” as recognized by the state of Oregon. George specializes in curriculum development and instructional improvement as well as working with at-risk students and closing the achievement gap. Connect with George on Twitter @runnin26:
Three words—timed pair share.
This is a simple easy to use strategy and much more effective than a traditional turn and talk. Don’t believe me? Just picture a time when you were sitting next to someone like me who likes to talk. My guess is that a question was posed and you were given time to talk to your shoulder partner. During the allotted time you had to listen to your partner go off passionately about the topic leaving you little, if any, time to respond with your own thoughts. Yep, that is turn and talk at its finest. A small tweak and totally underutilized structure is the timed pair share.
As many of you know, this is again done with a shoulder partner but, instead of hogging the entire talking time, you are given a specific amount that you get to talk—without interruptions. Worried that you can’t talk the entire time? Don’t worry. In case you finish early or don’t have much to say, your partner helps coach you with prompting questions. By creating a structured partnership, you provide equity of voice and your participation level increases significantly compared to just calling on students with equity sticks or hands in the air. We implemented this in our math classrooms this past year in the intermediate grades and saw a huge increase in student achievement and growth with our students from poverty, English Language Learners, and students with special needs.
So while it may sound simple, this teacher and student move yields high results and is often overlooked.
Response From Laura Greenstein
A lifelong educator, Dr. Laura Greenstein has served as a teacher and school leader, professor and professional development specialist. Her passion for excellence in assessment is evident in her numerous books, articles, and blogs on the topic:
Cultivating Assessment AS a Learning Maximizer
Assessment is something that comes at the end of teaching and learning: Right? Educators would most likely disagree and explain that assessment is more than a summative test or measure. Assidere, meaning to sit beside, is the Latin root of assessment and reflects its early use in determining the rate or merit of something. Assessment, rather than being done to students, means doing something with students to ascertain and monitor the pace or quality of learning or, even better, to engage students as efficacious assessors. And at best, to identify missteps and misconceptions that guide responses to student learning.
With more student focused and embedded assessment comes empowerment and accountability. When the assessment is the learning, students become increasingly metacognitive, increasing their awareness of how they learn, taking responsibility for learning, and regulating those processes. As a result, self-confidence, and independence increase. Ideas for translating theory into routines of best practice are shown below.
When assessment is seamlessly integrated throughout teaching and learning, these things happen:
- Assessment develops into a collaborative and mutual process
- Insights into learning are used by both student and teacher
- Assessment is visible, sticky, and transparent
Editor’s Note: The next three responses talk about the use of the Question Formulation Technique (QFT):
Response From Dan Rothstein
Dan Rothstein is Co-Director of the Right Question Institute:
There are many intriguing examples of innovative efforts to encourage personalized learning, project-based learning, inquiry-based learning, deeper learning, more deliberate scientific thinking models and more. There are also serious efforts to improve science teaching and learning through the Next Generation Science Standards and social studies instruction through the C3 Social Studies Framework. All of these are carefully designed to lead to students who are more engaged in their own learning, thinking more rigorously and critically about what they are learning and, eventually, can drive their own learning.
In the six years since the Question Formulation Technique (QFT) was introduced to educators via Make Just One Change: Teach Students to Ask Their Own Questions (Harvard Education Press: 2011), we have seen remarkable ways educators have used the QFT to transform their classrooms into hubs of activity, alive with students asking questions and driving their own inquiry. There is now vast expertise in best practices in teaching students to ask their own questions in schools and districts all around the country.
Here are two quick examples of how educators are driving adoption of the QFT in science and social studies as a way to support and highlight teacher-led innovation. Their enthusiasm for the QFT derives from its value as an Occam’s Razor solution—using the simplest, most effective model—for the challenging task of promoting student inquiry. When speaking about what may be under-appreciated, we need to recognize that teachers’ hard work is too often under-appreciated. They deserve teaching models and tools that make their jobs easier and bring greater joy to teaching and learning for both teachers and students.
Response From David Jacob
David Jacob is a curriculum designer and professional developer in the lower Hudson Valley region in New York. He is leading an elementary curriculum development project at PNW BOCES for regional school districts and provides professional development for all levels of science education. David has presented science curriculum development strategies at state and national science education conferences:
As many states are adopting new science standards like the Next Generation Science Standards (NGSS), teachers are struggling with how to effectively teach the Science and Engineering Practices (SEP), which is one of the three dimensions of these new standards. These Practices are foundational to the way in which scientists and engineers approach their work. One of the goals of these new science standards is that students be able to investigate science and engineering through inquiry-based tasks, not just by reading informational texts.
One of the SEPs that I found surprisingly challenging was “Asking Questions (for Science) and Defining Problems (for Engineering).” When I first started to implement this Practice with teachers during professional development, I was surprised by how much adult learners struggled to produce thoughtful or insightful questions about what they were learning or about the scientific phenomena they were shown. When a workshop participant started to form a question, we, as a group, would coach, support, and discuss the depth of the question. This process was productive but time-consuming and situational. My resulting question was, “Would teachers be able to help students produce similar results?”
Then a colleague introduced me to the Question Formulation Technique (QFT). This protocol was a simple yet effective way to not only produce questions quickly but also to engage students in metacognition, specifically regarding questioning and devising actionable next steps based on prioritized questions. I quickly tried this protocol on my next group of adult learners, and not only was the protocol simple to execute, but the questions that evolved demonstrated deep thinking around the Question Focus that I gave them (an exploding seed pod that dispersed seeds https://www.ngssphenomena.com/#/exploding-seeds/). I knew immediately that the QFT protocol was a perfect match to help students achieve at least one of the Practices of the NGSS.
As my team started to develop science curriculum for the second grade, we decided to infuse at least one lesson utilizing the QFT protocol into each of our units. We then had our 2nd grade teachers and curriculum writers test the protocol in their classrooms to get a sense of its appropriateness. We will be piloting this new science curriculum in about 24 classrooms in the Hudson Valley Region to gather more feedback. Our plan is to use the QFT protocol throughout the K-5 science curriculum to help students develop questions that will lead to inquiry-based investigations.
Response From Greg Brown
Dr. Gregory S. Brown is a curriculum and instructional specialist at PNW BOCES in New York’s lower Hudson Valley where he coordinates the Integrated Social Studies/ELA Curriculum and a Blended Learning Consortium of High Schools. Dr. Brown’s work focuses on building the capacity of teachers and leaders and he regularly speaks at conferences and provides professional development:
The development of the National C3 Framework and New York State’s adoption of the NYS Social Studies Framework in 2014 have left practitioners struggling to operationalize the four Dimensions of the C3 and the six Practices found in the NYS SS Framework. With emphasis being placed on Dimension 1 (Developing Questions and Planning Inquiries) and Practice A (Gathering, Interpreting, and Using Evidence), teachers are required to fundamentally shift their role from disseminators of information and content to inquiry guides. From a pedagogical standpoint, these shifts in instructional technique are supported by The Danielson Framework, Domain 3b (Using Questioning and Discussion Techniques), and Marzano’s Art and Science of Teaching, Design Question 4 (What will I do to help students generate and test hypotheses about new knowledge?). However, as is often the case, transferring educational theory into practice can be a challenge, and student generated questioning is no exception. This struggle has resulted in classrooms where inquiry is often designed around teachers asking students questions and students relying on resources provided by the teacher to answer them. Consequently, an important opportunity to leverage student interest, voice, and motivation is missed, as true inquiry is diminished and replaced with an exercise rooted in task completion.
This is where the Question Formulation Technique (QFT) comes into play. The QFT is an easy way for anyone to create and prioritize questions and then reflect on what they have learned (Convergent, Divergent, and Metacognition) before establishing next steps. I was introduced to the QFT at a statewide meeting of curriculum and staff developers last fall and have since had the opportunity to apply it to my work with students and teachers of all grade levels and content areas. The results have been astounding. Teachers are excited to turn over control to students, and students are empowered to take ownership of their learning. The QFT is an ideal way to build students’ capacities to question the world around them, find answers to questions that matter, and become more accountable for their learning. In short, the QFT has allowed me and my colleagues to better realize one of the core tenets of the C3 Social Studies Framework and thereby help our students become the engaged, curious, and informed citizens of the future we desire.
Thanks to Jo, Katie, Rachael, Laura, Dan, David, and Greg for their contributions!
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