Category Archives: Mathematics

And now for something completely different

Mathematics
Real life meets Geometry class…
A few months ago, my students (Sarah Hampton here) were able to design and build a parking lot for our school. In their own words, here’s how it happened.  This blog post was written by them.

Parking Lot: What’s the problem?
We have a huge real world problem that our geometry class can solve! Our school’s upper field parking area is somewhat of a mess. There are no instructions as to where parking is permitted, so, as a result, many drivers claim more parking space than needed and don’t leave any space for other drivers. This leads to a traffic jam, causing a slower and confusing flow of traffic. In addition, because of this catastrophe, many drivers are forced to drive on the running track in order to exit the area, thus damaging the surface and placing pedestrians at risk for being injured.

In order to address these problems, Mr. Mark Hill, the Head of the Building and Grounds Committee, tasked our Geometry class to design a parking lot. We had to fulfill the needs of a counterclockwise flow of traffic, follow local regulations, and maximize the number of parking spaces, all while making safety our number one priority. This fell into a two part project, first, we designed a blueprint for the parking lot, and secondly, we laid out the actual parking lot.

Our small class was divided into two teams: a team of the three girls and one of the four boys. To get to the best solution, the teams competed on making the best and most effective design possible. After working hard, both teams presented a pitch to three judges, Mrs. Hampton, our geometry teacher, Mr. Hill, the Head of Building and Grounds and a civil engineer, and Mr. Vermillion, our Head of School. As the pitch started, Mr. Hill set the tone for the students saying “Let me tell you this; this project is as real-world as it gets. If you were an engineering consulting firm, you would be doing the same thing right now. You would prepare a preliminary solution to the problem and “pitch” that to the project owners. In this case, that’s Sullins Academy. If we liked your design, we’d hire you to do the work. As students, you may get to see your design actually implemented, which will be a tangible reminder of your time here whenever you go up to the field.”

The three judges came to a conclusion that there were positive elements in both teams’ designs. As a result, there was a draw and Mr. Hill made a new blueprint combining ideas of both teams. On a cool day, the class went up to the track to start marking the parking lot. We built a curb stop template and an angled line template and took all of our other supplies: string, stakes, measuring tapes, a speed square, and a few sharpies. Then we measured out the correct angle and distances for each parking spot, which used our knowledge in geometry and basic math to figure out where to put everything.
         
Throughout this project, we learned how to use an engineer scale, create a blueprint, and include trigonometry in real life situations. Most importantly, we learned the significance of proportionality in similar figures. In the end, we realized how much work and math really go into constructing a parking lot!

Implementing Bootstrap: An Adventure in Algebra and Computer Science Integration

Computational Thinking, Computer Science, Mathematics, Written by: Sarah Hampton

By Sarah Hampton
​

In a former post, I wrote about a site I discovered while exploring the 2016 Stem for All Videohall called Bootstrap.

Bootstrap designs curricula that meaningfully integrate rigorous computer science concepts into more mainstream subjects such as math and science. Developed with the help of Brown, WPI, and Northeastern, Bootstrap has backing from several major players including Google, Microsoft, and the National Science Foundation. If that isn’t enough to pique your interest, initial research shows that Bootstrap is one of the only computer science curriculums that demonstrates measurable transfer to algebra, specifically on functions, variables, and word problems. (Wright, Rich, & Lee, 2013 and Schanzer, Fisler, Krishnamurthi, & Felleisen, 2015)

Recently at our school, Sullins Academy, the middle school math teachers (including myself) and the schoolwide technology teacher met to discuss and coordinate implementation of Bootstrap’s algebra curriculum for our eighth graders. The curriculum combines principles of mathematics and programming as students create their own simple video game. Before the meeting, we independently worked through the first unit which included dissecting the parts of a video game, relating the coordinate plane to positioning, relating the order of operations to program evaluation, and planning our own basic video game. After talking about our reactions to unit one, we worked through unit two, distinguishing data types used by programs and writing functions to manipulate them, as a group.

After working through the first two units, we knew Bootstrap was something we wanted to try with our students for three main reasons:

  1. Bootstrap makes algebra relevant and accessible to all learners. This could be a game-changer for traditionally disengaged math students.
  2. Computational thinking (CT) is huge for computer science and math, and Bootstrap is a great way to develop it. According to the Center for Computational Thinking at Carnegie Mellon, CT is “a way of solving problems, designing systems, and understanding human behavior that draws on concepts fundamental to computer science. To flourish in today’s world, computational thinking has to be a fundamental part of the way people think and understand the world.” We agree and want to actively cultivate CT in our students.
  3. Bootstrap might be more motivating for students than a block language like Scratch because they are typing real code. They might feel more as if they are engaged in “real” programming. (Although, we know that the learning outcomes of Scratch can be extremely high-level and beneficial, we have heard students make derogatory comments about block languages being elementary.)

So we knew we wanted to implement Bootstrap, but we still had a big question: when and through what class (math or technology) would this be taught? Similar to most cross-curricular projects, there would be difficulty meeting standards organically for both classes. We decided to implement the curriculum predominantly through the technology class with crossovers in the eighth grade math classes as they naturally arise. (I am lucky to work in a school where we are encouraged to work across classes. Flexibility and collaboration are two of my favorite things about our school.)

Now that we have a plan in place, we are all really excited about the potential learning outcomes. We hope it shows students that math and technology do not exist in individual bubbles and that standards are not just isolated facts to memorize or know for a test. All subjects and content are integrated in real life for authentic purposes. The technology teacher hopes that this will make students realize that programming is within their grasp. It’s not this abstract, crazy, no-way-I-can-do-it sort-of-thing thing. Even if students don’t program again, the technology teacher hopes that it helps with troubleshooting abilities and independence. In addition, she hopes it will motivate students to improve their typing skills and realize why attention to detail is important, for example, when they see that even one missing parenthesis or misspelled word will break the program. Beyond the obvious desire for students to better understand algebra, the math teachers hope it allows students to see that math is really useful beyond the classroom. Most importantly, we hope working on Bootstrap displaces the teacher and puts the students at the center of the learning by improving metacognition and developing perseverance as they work through their error messages. In this way, students might grow out of the teacher-dependent mentality and learn to trust and rely on themselves and each other.

Keeping it real, we are concerned about a few things as well. It was interesting to see our reactions to the curriculum because the technology teacher has ample programming experience, I only have some, and the third teacher has no former experience. This was a fortunate coincidence because it represents the spectrum of prior knowledge our students will have as well. Overall, Bootstrap provides enough scaffolding for any previous exposure to programming as long as you are comfortable with a “learn as you go” approach, although occasionally, it did seem as if Bootstrap made an optimistic assumption about what students would know coming in. For those with no prior experience, we would have liked more direct instruction on key vocabulary, syntax requirements, and reading and diagnosing error messages. Another concern is keeping all students engaged for the length of the project. Undoubtedly, some students will be able to fly through the curriculum while others need a bit more time. We hope the answer to this problem lies in offering the extensions Bootstrap has built in for quick learners.

Overall, we are really looking forward to seeing what Bootstrap can do for our students. Our plan is in place so may the adventure continue! I will keep you posted.

Have any of you implemented Bootstrap or another computer science curriculum like Logo or Scratch? Did you see transfer to math or science? What advantages did you notice? Are there any obstacles you can help us navigate? We would love to learn from you!

Citations and Further Reading
Schanzer, E., Fisler, K., Krishnamurthi, S., & Felleisen, M. (2015). Transferring Skills at Solving Word Problems from Computing to Algebra Through Bootstrap, ACM Technical Symposium on Computer Science Education, 2015.
Available at:
http://cs.brown.edu/~sk/Publications/Papers/Published/sfkf-trans-word-prob-comp-alg-bs/paper.pdf

Wright, G., Rich, P. & Lee, R. (2013). The Influence of Teaching Programming on Learning Mathematics. In R. McBride & M. Searson (Eds.), Proceedings of SITE 2013–Society for Information Technology & Teacher Education International Conference (pp. 4612-4615). New Orleans, Louisiana, United States: Association for the Advancement of Computing in Education (AACE).
Available at:
https://www.learntechlib.org/p/48851/.

Center for Computational Thinking at Carnegie Mellon.
Available at:
​https://www.cs.cmu.edu/~CompThink/

Education Wonderland: STEM for All Video Showcase

Mathematics, Video Showcase, Written by: Sarah Hampton
​By Sarah Hampton

I wish there was an extra planning block built into every teacher’s day for locating quality, relevant resources. Educators and researchers are out there doing amazing things that I rarely hear about through the grapevine. Yet, when I spend a bit of time down rabbit holes on the internet, I stumble across exciting and innovative practices like STEP: Science through Technology Enhanced Play in which young students pretend to be bees and watch their bees interact on screen while an XBOX Kinect sensor bar maps their movements. If you have had similar challenges finding resources, then I have GREAT NEWS for you! Researchers funded by the National Science Foundation have created three-minute videos of some of the best things happening in STEM education in their projects and share them in a showcase. I have watched most from last year’s showcase, and I was surprised to see how many were free, easily implementable, and relevant across all disciplines–even those not traditionally considered to be under the STEM umbrella such as geography. You can also filter the videos by subject or grade level to find ones most helpful to your classroom.

As a science teacher, there are several hands-on activities that easily correlate to the content. As a math teacher, meaningful, engaging opportunities are harder to find. That’s why I was thrilled when I saw this video on teaching Algebra through coding using Bootstrap. The connections to Cartesian coordinates, the distance formula, and functions are tangible as students create their own video games. I have already proposed this idea to another math teacher and tech teacher at my school and they have responded with enthusiasm and buy in. We are hoping to meet over the summer to work through the free curriculum ourselves with intent to implement it through the eighth grade technology class next year.

My trip down this particular rabbit hole felt so much like Wonderland that I am counting down the days until the 2017 Stem for All Video Showcase: Research and Design for Impact funded by NSF beginning May 15. I hope to find you there. More importantly, I hope you find resources to implement in your school there. This is an exciting time to be in education! Check out the showcase and find out why!

Picture

​Students pretend to be bees in STEP.  STEP uses OpenPTrack, an open source platform for sensing position and movement of large groups of people.  

Students write basic code to program their own video games in Bootstrap as a means of learning algebra.

Favorite Tech Tools Series: Google Drive

Classroom Technology, Collaboration, Mathematics, Written by: Sarah Hampton
Edited 2/11/2018 to add the link to the gold award!  Congratulations, Sarah!

Picture


By Sarah Hampton
​
From STEM programs to one-to-one device campaigns, we hear a lot about the importance of technology in the classroom. Like most initiatives, this is for good reason! We live in the digital age, and producing students who can responsibly and productively use the numerous technologies at their disposal is a crucial 21st century skill. Also like most initiatives, our tendency might be to view technology use as a bothersome requirement handed down by well-meaning administrators. When we approach anything with this attitude (read: the oft-dreaded professional development), we miss out on the spirit of the requirement. In this case, that means implementing technology in ways that genuinely improve student learning or enhance classroom organization and workflow. In this series of posts, I will share my favorite tech tools for streamlining my middle school classroom and promoting student learning. Let’s start with Google Drive, one of my favorite student-centered learning tools.

Google Drive
Technology is useful when it allows you to do something you can’t do with a whiteboard and markers, or when it allows you to do something better or faster. Google Drive frequently allows me to do both. You probably already know that Google Docs is a powerful collaborative writing tool. Multiple studies have found that web-based collaborative activities, done well, can promote learning outcomes, teamwork, social skills, and basic computing skills among students (Zhou, Simpson, & Domizi, 2012, pg. 359-360). In addition, I love how easy it is to give comments in Google Docs and how easy it is for students to work together. If you haven’t incorporated it yet, then make a class writing project a priority. Here is one example. If you are already a Google Docs pro, then check into using Slides or Forms. Our school frequently uses Forms for quick polls and surveys. Google Sheets is also a must have, particularly for math and science teachers. I would like to demonstrate how powerful this app can be by sharing how it helped me create one of my best lessons this year for middle school algebra (my class included mixed ages of 6th, 7th, and 8th grade Algebra 1 students).

After watching the Olympics this summer, I started to wonder why some countries seemed to do better than others. I posed that question to my students and we brainstormed two main categories that we thought might correlate with a country’s Olympic performance: population (greater probability that gifted athletes live there) and per capita income (more opportunities for athletes to practice and/or have access to high quality facilities and equipment.) I had each student pick three to five countries, research their populations, per capita incomes, and total medal counts in the past four summer Olympics, and add their information to the class spreadsheet. Then, in groups, they created a scatterplot for their assigned factor and analyzed the data using linear regressions to see which factors more highly correlated with Olympic performance. If you want more specifics or want to see the results, then check out our class spreadsheet. You can also find instructions for a similar project at Mathalicious.

This project was organically cross-curricular and addressed multiple algebra standards by necessity. It incorporated geography, because the students placed push pins in their countries on a giant world map, and economics, because they wondered why some countries’ per capita incomes were very high or very low. It gave meaning to population density when the students saw the size of a country on the map and then noted its population on the bar graph. (Iraq and Canada have similar populations? But Canada is soooo much bigger!) It increased number sense when they created bar graphs, scatterplots, and histograms and realized that some of the values were literally off the charts–like the per capita income of Monaco (which presented the perfect opportunity for me to introduce vocabulary like “outlier.”) Astonished, students were naturally curious enough to research why. This led to lessons on digital literacy as we discussed how to appropriately locate, evaluate, and use information from the internet, a skill that is frequently overestimated in today’s students according to a study commissioned by the British Library and JISC (University College London, 2008).

The students really got into this project and even asked to do an extension! They hypothesized that countries with lower average temperatures would perform better in the winter Olympics, so, of course, we analyzed that, too. This matches perfectly with the International Society for Technology and Education’s claim that, “When students take responsibility for their own learning, they become explorers capable of leveraging their curiosity to solve real-world problems” (ISTE, 2017).

As it turns out, we weren’t the only people to look at what factors affect Olympic performance. After the project, my students found two websites that helped explain things further. The first was written by an economics doctoral student and the second by a senior editor at The Atlantic.  (Bian 2005, O’Brien 2012) The other sites concluded that the same factors we studied were major contributors, and their charts and methods remarkably resembled our own, albeit with some more advanced statistics in the case of the doctoral student’s article. My students’ excited comments indicated that they felt validated in their reasoning and felt that they were doing “real math.”

This project hit the sweet spot: students were engaged in deep and relevant learning, and Google Sheets significantly contributed to its effectiveness.

​How have you used Google Drive to create more student-centered environments? What outcomes did you see when you used them? Did anything (good or bad) surprise you? I would love to learn from your experiences by reading your comments!

Picture

Students proudly displayed their results in the hallway outside our classroom.
Citations and Further Reading
Bian, X. (2005). Predicting Olympic Medal Counts: the Effects of Economic Development on Olympic Performance. The Park Place Economist, 13(1), 37-44. Available at: https://www.iwu.edu/economics/PPE13/bian.pdf

International Society for Technology and Education. (2017). Essential Conditions: Student-Centered Learning. Available at: http://www.iste.org/standards/tools-resources/essential-conditions/student-centered-learning

Mathalicious. (2017). Hitting the Slopes. Available at: http://www.mathalicious.com/lessons/hitting-the-slopes

National Writing Project. (2017). Directions for Teachers Participating in Letters to the Next President: Writing Our Future. Available at: http://www.nwp.org/cs/public/print/doc/nwpsites/writing_our_future/directions.csp

O’Brien, M. (2012). Medal-Count Economics: What Factors Explain the Olympics’ Biggest Winners? The Atlantic. Available at: https://www.theatlantic.com/business/archive/2012/08/medal-count-economics-what-factors-explain-the-olympics-biggest-winners/260951/

University College London. (2008). Information Behaviour of the Researcher of the Future. Available at: https://www.webarchive.org.uk/wayback/archive/20140614113419/http://www.jisc.ac.uk/media/documents/programmes/reppres/gg_final_keynote_11012008.pdf

Zhou, W., Simpson, E., & Domizi, D.P. (2012). Google Docs in an Out-of-Class Collaborative Writing Activity. Journal of Teaching and Learning in Higher Education, 24(3), 359-375. Available at: http://files.eric.ed.gov/fulltext/EJ1000688.pdf

The Benefits and Obstacles of Constructivism

Constructivism, Mathematics, Science, Written by: Sarah Hampton

Picture

By Sarah Hampton

Sarah Hampton teaches middle school math and science at Sullins Academy in Southwest Virginia.  She has ten years of teaching experience in various disciplines and settings.

Over the last sixty years, thousands of articles have looked at whether or not constructivism works.  I wanted to understand the research, but it was overwhelming.  However, thanks to advances in technology and fancy statistics, researchers can analyze aggregate data on the subject.  After reading multiple articles and three meta-analyses (an analysis that aggregates data and allows you to look across many studies) specifically regarding science education and constructivism, two things became apparent.  First, it is extremely difficult to show an impact of instructional strategies on student learning outcomes!  Out of 1500 studies in one analysis, only six of the studies met the criteria that allow causal inferences to be made (Furtak, Seidel, Iverson, & Briggs, 2009, p. 27).  Second, despite that difficulty, the evidence favors constructivism.  The conclusions from all three meta-analyses demonstrated statistically significant positive effects of constructivist practices on student learning.  So, if we know constructivism is good for our students, then why do we not see more of it in action?  To hit a little closer to home, if I know these are good practices, then why am I not doing more of them?  I think there are some legitimate obstacles.  Here are my top three:

Obstacle 1:  The time it takes to find or create relevant, quality tasks
The number of daily teaching requirements and professional demands apart from planning are enough to fill our workday!  Planning inquiry instruction is extremely time-consuming because you have to sort through all of the activities that aren’t that great or don’t apply to your subject or grade level. Half the time I end up creating my own from scratch, which is also a time drain. In contrast, planning for direct instruction is a snap. Decide what you want to cover and write down the topic in your lesson plans.  Done.  As a result, to save time, we often revert to direct instruction (otherwise we cut into our family time to plan).

Proposed Solution A:  Find a resource that produces quality learner-centered, constructivist materials and start there.  For math, I use http://www.mathalicious.com/ and https://illuminations.nctm.org/.  Both allow you to filter by topic and grade level, which saves additional time.  For science, I like http://www.middleschoolchemistry.com/.  

Proposed Solution B:  Try to view the time spent on finding quality materials as a necessary startup cost.  If you like them, then you can recycle them year to year.  In addition, Berland, Baker, and Blikstein argued that constructivism can actually save time when fully implemented by enhancing “classroom dynamics that may streamline class preparation (e.g., peer teaching or peer feedback)” (Berland, Baker, & Blikstein, 2014).

Obstacle 2:  The instructional time it requires to implement meaningful tasks
I don’t know about you, but I start my year feeling behind!  There just doesn’t seem to be enough time for my students to deeply comprehend the required algebra or physical science concepts as dictated by state and national standards within the given time frame.  When we allow the pressure of the standards and test to dictate our instructional practices, we begin to look for the fastest possible way to disseminate information, and direct instruction is efficient—we just tell them what it is we want them to know.  However, efficient is only efficient if it is also genuinely effective.  

Proposed Solution:  Try to see beyond the standards and the test.  D.F. Halpern expressed concern about our preoccupation with these and said, “We only care about student performance in school because we believe that it predicts what students will remember and do when they are somewhere else at some other time.  Yet we often teach and test as though the underlying rationale for education were to improve student performance in school.  As a consequence, we rarely assess student learning in the context or at the time for which we are teaching” (Halpern & Hakel, 2003, p. 38).  

I am not a teacher because I want my students to pass a test.  I am a teacher because I want my students to excel in life.  Constructivist practices require students to think critically and creatively, innovatively problem solve, collaborate, and communicate—therefore preparing students for the test and beyond.  As Hmelo-Silver, Duncan, and Chinn (2007) argued, “This evidence suggests that these approaches can foster deep and meaningful learning as well as significant gains in student achievement on standardized tests” (p. 99).  I suspect the greatest benefits of constructivism are immeasurable and consequently undocumented and marginalized.  I would love to know the impact on long-term retention, higher order thinking, lifelong learning, and employer satisfaction.  

Obstacle 3:  The difficulty of meshing inquiry and explicit instruction
I want my students to do the work of the learning, so it doesn’t seem like inquiry if I’m leading the discussion. But sometimes whole group instruction makes the most sense for the instructional goal.

Proposed Solution:  Adjust your understanding: constructivism does not preclude explicit instruction.  You are probably engaged in more constructivism during whole group instruction than you think.  Simple strategies like accountable talk and purposeful questioning lead to minds-on learning even when students aren’t engaged in hands-on learning (Goldman, 2014).   Constructivism is often equated with minimally guided instruction, but they are not synonymous.  In fact, “most proponents of IL (inquiry learning, a type of constructivism) are in favor of structured guidance in an environment that affords choice, hands-on and minds-on experiences, and rich student collaborations” (Hmelo-Silver et al., 2007, p. 104, emphasis added).  

The goal of constructivism is for our students to actively construct meaning for new information rather than passively accepting our word for it.  Since we can create opportunities for our students to do this in multiple ways, we should focus on the culture of constructivism rather than the day to day teaching methods we use to maintain that culture.

In conclusion, constructivism isn’t easy, but it is necessary to help students learn.  It’s worth finding a way to overcome the obstacles.  If you are interested in reading more about why, then please see below for a complete list of the works I cited and consulted.  Don’t forget to leave your own comments – I would love to hear your obstacles and solutions, too!

Citations and Further Reading 

Alfieri, L., Brooks, P. J., Aldrich, N. & Tenenbaum H. R. (2011). Does discovery-based instruction enhance learning?  Journal of Educational Psychology, 103(1), 1-18.  
Available at: http://www.cideronline.org/podcasts/pdf/1.pdf

Berland, M., Baker, R. S., & Blikstein, P. (2014). Educational data mining and learning analytics:
Applications to constructionist research. Technology, Knowledge and Learning, 19(1-2),
205-220.
Available at: https://pdfs.semanticscholar.org/41c0/0af6ce63b919530ea691d058e8725d33d901.pdf

Furtak, E. M., Seidel, T., Iverson, H., & Briggs, D. (2009).  Recent experimental studies of
inquiry-based teaching: a meta-analysis and review, European Association for
Research on Learning and Instruction, Amsterdam, Netherlands, August 25-29, 2009.
Available at: http://spot.colorado.edu/~furtake/Furtak_et_al_EARLI2009_Meta-Analysis.pdf

Goldman, P.  (2014, January 22). #2. What is Accountable Talk®?  Institute for Learning
Podcast.
Available at: http://ifl.pitt.edu/index.php/educator_resources/accountable_talk/podcasts/2

Halpern, D. F. & Hakel, M. D. (2003). Applying the science of learning to the university and
beyond: teaching for long-term retention and transfer.  Change, July/August 2003,
36-41. Available at: http://www.chabotcollege.edu/planning/outcome-assessment/index.php

Hmelo-Silver, C. E., Duncan, R. G. & Chinn, C. A. (2007).  Scaffolding and achievement in
problem-based and inquiry learning:  a response to Kirschner, Sweller, and Clark
(2006).  Educational Psychologist, 42(2), 99-107.
Available at:
http://www.sfu.ca/~jcnesbit/EDUC220/ThinkPaper/HmeloSilverDuncan2007.pdf

Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction
does not work: an analysis of the failure of constructivist, discovery, problem-based,
experiential, and inquiry-based teaching.  Educational Psychologist, 41(2), 75-86.  
Available at: http://cogtech.usc.edu/publications/kirschner_Sweller_Clark.pdf

Lang, Albert. (2010).  Executives Say the 21st Century Requires More Skilled Workers.  
Available at:
http://www.p21.org/news-events/press-releases/923-executives-say-the-21st-century-req
uires-more-skilled-workers

Minner, D. D., Levy, A. J., & Century, J.  (2010).  Inquiry-based science instruction – what is it
and does it matter?  Results from a research synthesis years 1984 to 2002.  Journal of Research in Science Teaching, 47(4), 474-496.
Available at:
https://www.ntnu.no/wiki/download/attachments/8324914/JRST-Inquiry-based+science+instruction+-+what+is+it+and+does+it+matter-+Results+from+a+research+synthesis+years+1984+to+2002.pdf

Schroeder, C. M., Scott, T. P., Tolson, H., Huang, T., & Lee, Y. (2007).  A meta-analysis of
national research:  Effects of teaching strategies on student achievement in science
in the United States.  Journal of Research in Science Teaching, 44(10), 1436-1460.  
Available at: http://cudc.uqam.ca/publication/ref/12context.pdf

Shah, I. & Rahat, T. (2014). Effect of activity based teaching method in science.  International
Journal of Humanities and Management Sciences, 2(1), 39-41.  Retrieved from
http://www.isaet.org/images/extraimages/K314003.pdf

Stohr-Hunt, P. M. (1996). An analysis of frequency of hands-on experience and science
achievement.  Journal of Research in Science Teaching, 33(1), 101-109.
Available at: https://vista.gmu.edu/assets/docs/vista/JournalOfResearch.pdf

Windschitl, M. (1999).  The challenges of sustaining a constructivist classroom culture.  Phi
Delta Kappan, 80(10), 751-756.
Available at: http://www-tc.pbs.org/teacherline/courses/inst335/docs/inst335_windschitl.pdf?cc=tlredir

Active Learning Day, 2016

Constructivism, Mathematics, Science, STEM, Written by: Judi Fusco
By Judi Fusco 

Active Learning Day is Today, October 25!  What are you doing for it? What will active learning look like in your classroom? In active learning, students work on meaningful problems and activities to help them construct their learning. This includes inquiry activities, discussion and argumentation, making, solving problems, design, and questions.

Last month, we had the pleasure of helping organize the Active Learning in STEM Education Symposium, sponsored by NSF as part of the activities honoring the Presidential Awards for Excellence in Mathematics and Science Teaching awardees. The keynote speaker, Bill Penuel, focused on “talk” — particularly “accountable talk” — as an important activity to support Active Learning. 

If you want to know more about accountable talk, take a look at the Talk Science Primer by TERC. There are many great tips for teachers of all subjects in there. For Math Classrooms, here’s a link discussing Creating Math Talk Communities. For general information about it see ASCD’s Procedures for Classroom Talk.  
​
In the Active Learning in STEM Education Symposium, one of the presenters, Joe Krajcik, discussed Interactions, a curriculum aligned with the Next Generation Science Standards (NGSS) to make science an active endeavor in a classroom.  (Visit the Interactions project page and click on the curriculum tab to learn more.) Language and talk are essential in NGSS. You may want to check out the videos on the NSTA site where you can see what NGSS looks like in action. You can also see what NGSS looks like in a 4th grade Science Classroom; this video was shown in the Active Learning Day in STEM symposium by Okhee Lee as she discussed NGSS for all Students including English Learners.  

Other presentations at the symposium included Jennifer Knudsen on Bridging Professional Development and the idea of using Improv in a Math class, Eric Hamilton on collaborating with a cyber-ensemble of tools, Tamara Moore on using mathematical modeling to engage learners in meaningful problem solving skills, David Webb on AgentCubes as active learning, and Nichole Pinkard on Digital Youth Divas and making eCards to learn about circuitry.  (See links to the presentations of all the speakers on the site. ) 
Active Learning Day is officially today, but there’s no reason why you can’t do more in your classroom at any time.  Leave a comment and tell us about what active learning looks like in your classroom!

Practitioner POV of Constructivist Approaches

Constructivism, Mathematics, Written by: Pati Ruiz

Picture

By Pati Ruiz

Pati Ruiz is a doctoral student in Learning Technologies at Pepperdine University. She has worked as a teacher (Spanish and Computer Science), Director of Learning Technology, and is the incoming Dean of Studies at Convent of the Sacred Heart in New York City.

​
As a teacher studying the learning sciences in graduate school, I understood  constructivist practices in theory, but I often wondered what constructivism looked like in action. Taking a constructivist perspective, Windschitl (2002) describes learning as an act of both individual interpretation and negotiation with others, where knowledge is the collection of what is constructed individually and collectively. In classrooms, a constructivist or open approach should support learners in actively constructing their own knowledge, but what does that really look like? How much time does it take? What are the challenges?

I spoke with some teachers to learn more about how they support an open approach to learning in their classrooms. This post will focus on the strategies of a middle and high school math teacher I interviewed. Future posts will focus on the work of high school Spanish and English teachers.

Math Classroom Example: Christine Trying a New Curriculum
Christine DeHaven is in her fifth year teaching middle and high school math. In her Honors Algebra 1 class at Pacific Ridge School, she always taught in a very guided or instructivist approach. For example, if the topic was lines, she would first lecture about lines, then work out one example in front of the class, and then students would do some problems on their own in class. For homework, students would complete more problems that progressed in difficulty. The class would then move on to the next topic. Christine noticed that students were just memorizing steps instead of problem solving, so she decided she needed to change her teaching approach.

Christine had learned about a new curriculum that allowed the students to learn through conferences and visits to other schools, including The Bishop’s School, Deerfield Academy, and Phillips Exeter Academy (where the curriculum was developed). Christine and a colleague decided to swap their traditional direct-instruction approach for a problem-based approach. Christine had seen the new curriculum in action, and felt that it could work at her school, too. Still, she modified the curriculum slightly for her students. Sometimes the transition to a new approach needs to be done gently. Here is what Christine’s curriculum looks like now.

First, students are assigned 8-10 homework problems per night. The goal for students is that they attempt all of the problems before class. When they arrive to class the next day, students pick a problem to solve on the board. Multiple students may put up the same problem, and everyone contributes at least one problem. After all of the problems are on the board, groups of students go to the board to present one problem at a time. If there are multiple solutions to the same problem, Christine leads a discussion about which solution is more efficient. With this new method, Christine finds that her students have more ownership of what they are learning. They apply problem solving skills to the homework and construct their own understandings through their solutions and conversations about their solutions. When they present their work and discuss the various solutions, students gain a better understanding of the concepts because they have to make a case for or against a certain way of solving a problem. Christine also encourages students’ use of graphing as a method to solve the homework problems. Students use tools like the Desmos Graphing Calculator to see a visual representation of the problem. In this way, Christine guides students to look at problems in three different ways: numerically, algebraically, and visually.

Parent education and administrator support has played an important role in the ease of adopting her new curriculum. While Christine initially received some negative feedback about her approach from parents, she felt well supported by her school administrators who are able to point concerned parents towards research and articles about the success of this approach. Open house became an opportunity for Christine to educate concerned parents–she even encouraged them to work with their children to solve the daily homework problems. Christine still attempts to engage parents by encouraging them to follow along on the course website. Many do, and often share stories of working on problems with their children. While parents were initially skeptical, many now tell Christine how much they appreciate the new approach and they have fun helping their children with their math homework. In the beginning, Christine also got negative feedback from students. But – for the most part – they have come around now that they have more practice with the approach. Something else that has helped students adjust is that the homework problems they are solving are very realistic; students can relate to them. For example, one problem, which aims to help students understand how dangerous glancing at a phone is when driving, asks students to compute how far they would drive down the highway in the time it took them to read or respond to a text message. Many of Christine’s students are learning to drive or have friends who are, so problems like these are relevant and engaging to them. (Please don’t text and drive!)

Though challenging, Christine persisted in adopting the open curriculum because she felt that it was the best approach for her students. She thinks that students have a better understanding of the concepts they have covered. For example, they understand how to factor a polynomial and aren’t just guessing and checking. She reports they are able to prove why the square root of 2 is irrational. They also have a better sense of how a graph relates to algebra, and they persist in solving problems. When solving homework problems, students don’t always know the math theories or strategies they are using, but they are developing algorithms and figuring out problems as they go. These are essential skills for mathematics. Additionally, when students don’t solve a problem the first time, they are willing to try again and again. In this way, they are developing a growth mindset and starting to see the payoffs.

This approach has been more time consuming for Christine. It’s the first time she’s seen many of these problems on the homework, so she needs to solve them all in multiple ways before going to class. She needs to think like her students and try to anticipate the problems they’ll have and the misconceptions they might bring to a problem. This means she really needs to know the content she’s teaching. It’s more prep time before class, especially in the first year, but this way she knows how to guide discussions and ask the right questions. Christine uses her expertise to help students gain a deeper understanding and make connections to content they have seen before. She’s not lecturing as much anymore, but she remains the content area expert.

This idea leads to something that might be a struggle for some. It is described by Harland (2003) like this:
“When students arrived at a position where they could function well together and drive the enquiry forward, they seldom asked for help, and the teaching team no longer had their old roles and familiar student contact. Paradoxically, we felt some sense of loss at this stage and concluded that a lot of pleasure in teaching had gone…”

For Christine, though, she simply sees her role as a teacher changing. She is now more of a facilitator who ensures that students hit certain key points. She guides students in thinking more deeply by helping them ask questions instead of giving them answers. Her connection with students is now stronger, in her opinion. Preparing for class is more involved and time-consuming and her role in the classroom is smaller. But for Christine, that’s okay. What excites her about teaching is helping students discover the math that she loves, and she’s doing that.

If you’re interested in learning more about open approaches to Mathematics education, Christine recommended the Exeter Mathematics Institute and the Mathematics Visionary Project. We would love to hear what you think and the questions you might have for Christine or other teachers.

Harland, T. (2003). Vygotsky’s zone of proximal development and problem-based learning: Linking a theoretical concept with practice through action research. Teaching in higher education, 8(2), 263-272.

Windschitl, M. (2002). Framing constructivism in practice as the negotiation of dilemmas: An analysis of the conceptual, pedagogical, cultural, and political challenges facing teachers. Review of educational research, 72(2), 131-175.

Learning Scientists and Classroom Practice

Constructivism, Mathematics, Written by: Judi Fusco
​By Judi Fusco

As I promised in the previous post, here’s a look at Tesha Sengupta-Irving and Noel Enyedy’s 2015 article. In this post, I want to take a closer look at one study that shows the kind of work learning scientists do in classrooms with teachers. 

Some teachers (and principals, parents, and others) question whether student-driven (open) pedagogies work for students; they worry if students are on their own, they might waste valuable instructional minutes, especially in math classes. However, by exploring data, discussing and debating, and constructing their own understanding, students in an student-driven, open instructional approach achieve the instructional goals of the course as well as students in a teacher-led (guided or instructivist) approach. In addition, and importantly, students seem to enjoy learning mathematics more when taught with an open or constructivist approach versus a guided approach. In their article, Sengupta-Irving and Enyedy (2015) discuss how important enjoyment is in learning, and why and how a student-driven instructional approach helps them learn.

In the study, students’ test performance was the same for both the teacher-led and student-driven approaches. So why don’t we just stick with teacher-led techniques? Why do we want to switch to more student-driven approaches? Sengupta-Irving and Enyedy, and many other learning scientists, don’t think it’s enough to create mathematically proficient students without helping them develop an interest (or even love) for the subject that the student-driven approach helps create. Learning without enjoyment seems like a lost opportunity that may prevent students from doing well in the future. The authors think if students learn and enjoy subjects, those students might want to go further in the subject and take more classes.  

Using Learning Science as the Foundation to Build Practical Classroom Practices
So what did the students in the student-driven condition do while learning? On their own, the students started with a discussion to explore the data, tried to understand the problem, and debated the approach or solution with peers. They also experimented and during their discussion “invented” an understanding, in this case, of statistics. They (hopefully) invent what the teacher would have told them during a lecture. While it may seem inefficient to let students invent, because, after all, we could just tell them what they need to know, but the discussion and inventing engages them, helps them enjoy the subject, and strengthens their learning.

After they have gained some understanding on their own in their discussion, the teacher has a discussion with the students and helps them learn formal terms. Exploring first contrasts to what students do in the the instructivist or guided condition where the teacher tells them the formal terms, a great deal of information about the problem, what the important concepts are, and the approaches they should take in solving the problem. In the guided condition, students are not given an opportunity to explore informally.

For a long time, learning scientists have known that “telling” students after they have the opportunity to explore and develop their own understanding is more effective than telling them before they have had that opportunity (Schwartz & Bransford, 1998). Sengupta-Irving and Enyedy employ this learning science principle and find that students do well and seem to enjoy the lesson more. 

One other issue that is sometimes discussed about student-driven approaches is whether students are off-task when on their own. It is true, student-driven classrooms are usually noisier than instructivist ones, but that’s because there is learning occurring—in my experience, I have found that learning is a slightly noisy phenomenon. The researchers looked at off-task behavior in the two instructional approaches in the study and there wasn’t a difference. They found more instances of off-task behavior in the teacher-led condition than in the student-driven condition and approximately the same number of minutes of off-task behaviors in the two conditions. I think it’s important to note that the teacher in this research reported that she was more comfortable with the teacher-led approach. Because of that, the teacher may not have used an open approach very often, and her students may not have been as familiar with an open approach–yet there was no extra off-task behavior. To alleviate concerns that student-driven approaches require more time to work, both instructional approaches used the same amount of time for the lesson.

I want to go back to the issue of enjoyment. If, after a lesson, students don’t want to think about it any more—because it’s boring, one of the terms the students in the teacher-led condition used to describe the lesson—then we probably have not done the best we can for the students. Sure, if we tell students about something, we’ve gotten through the lesson and are able to cross that topic off the list. But shouldn’t learning be something more than just an item on a checklist? What if learning was enjoyable and students left wanting more? Learn the same amount, in the same amount of time, with very little off-task behavior, and enjoy it = win-win-win-win. And, add the bonus that enjoyment can potentially help students in their future work and motivate them to continue their studies. I’d make time for that in my classroom.

I’d love to know what you think about the article and their findings. In future posts, we’ll talk about how to o student-driven approaches and hear from teachers who have some good tips. I’d also love to hear how you teach and what you’ve seen or experienced in your classroom. Below you can read more details of the study.

Sengupta-Irving, T., & Enyedy, N. (2015). Why engaging in mathematical practices may explain stronger outcomes in affect and engagement: Comparing student-driven with highly guided inquiry. Journal of the Learning Sciences, 24(4), 550-592, DOI: 10.1080/10508406.2014.928214.

Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and instruction, 16(4), 475-5223.

​

Details of the study
In the study, one 5th grade classroom teacher taught two sets of students the same mathematics topic, for the same amount of time, using two different approaches: open (student-driven; 27 students) and guided  (instructivist; 25 students). The teacher was more comfortable with the guided approach, but had learned how to facilitate the open method and taught one class of students that way. The data collected included written assessments of the student’s work (a test), a survey inquiring about the students’ affect during the lessons, and video of the 5 hours of class time devoted to the topic for each instructional approach. The researchers report three main findings based on the analysis of this data:

  1. Assessment data showed that when students were given the opportunity to explore and solve problems in an open way working with their peers, they performed just as well as students who were in the guided (instructivist) situation. 
  2. Survey responses indicated that students in the open condition enjoyed the lesson significantly more, compared to guided students. Also, students in the open condition did not express any negative affect statements, but guided students did. (“Bored” was one of the negative affect statements used by the guided students.)
  3. Video analysis showed that in the two conditions, the amount of time spent in interactions between teacher and students, and students working together, were very similar. For example, for both conditions, there was a little over 3 hours spent in whole class activity and about 2 hours spent in small group work; during the small group work, adults spent about 1.5 hours helping the students with the lesson or managing behavior. Off-task time was roughly equal in the two conditions: there were 18 off-task instances (involving approximately 11 minutes (out of 300 minutes) of adult intervention) for off-task behavior in the guided condition, and 14 off-task instances (involving approximately 13 minutes (out of 300) of adult intervention) in the open condition.