Mathematics Updates
April 2017
Do You See and Engage Your Hidden Figures?
Excerpts from article by Matt Larson, NCTM President
By now most of us with an interest in mathematics or mathematics education have seen the powerful movie Hidden Figures — many of us likely more than once. The inspiring film focuses on the critical role of three African American women and their significant contributions to the mathematical and engineering work necessary to the initial success of the U.S. space program in the 1960s.
While the women in Hidden Figures were hidden, it clearly illustrates the postulate that mathematical talent is distributed across race, gender, and socio-economic status. At the same time, it painfully reminds us of the many educational, social, and professional opportunities denied African American women (and men) in the 1960s. Mary Jackson, a few years after Brown vs. Board of Education, still had to petition the court simply to attend night classes in engineering at the local high school because it was a segregated school.
We might like to think that access to upper-level mathematics courses is no longer an issue, but it is. Students from marginalized groups have less access to highly qualified mathematics teachers and less access to college preparatory pathways in mathematics (Nasir, 2016). Similarly, a recent report by the OECD (2016) found that more than 70 percent of students attend schools where the principal reports that students are grouped by “ability” for mathematics instruction. “Who teaches whom what?” remains a serious concern in K–12 mathematics in the United States.
Because structural obstacles remain in place in too many of our schools, we are still denying our society solutions to the many problems we face when we systematically ignore vast human potential. This is not to argue that the value of mathematics exists only to promote economic, defense, or scientific advancement, but its connection to each is clear. Francis Su, past president of the Mathematical Association of America, has argued that mathematics is ultimately for human flourishing — that it helps each and every one of us experience a well-lived life — whether or not we become professional mathematicians.
Hidden Figures also reminds us that education should instructionally emphasize collaboration, creativity, communication, problem solving, and innovation when it dramatically illustrates how the human computing groups were replaced by one of the first IBM mainframe computers. Yes, students need procedural fluency and conceptual understanding, but Dorothy Vaughan’s group would have lost their jobs had they not adapted, learned new skills, and been effective problem solvers — in the 1960s. The premium on continual learning and adaptation in the workplace has continued through the beginning of 21st century and is likely only to accelerate.
The movie also uncovers for many of us the significant mathematical contributions many African American women made, beginning in 1943, to aeronautical research as part of the West Computing Group at Langley, where they manually carried out complex computations for flight and space research as “human computers.”
For me one of the ways we can honor the pioneering work and contributions of Katherine Johnson, Dorothy Vaughan, Mary Jackson, and the entire West Computing Group is to not let the lessons of their life story fade from our consciousness. I encourage each of you to reflect on and discuss the following in your collaborative teams:
- Do you see mathematical potential in all your students, no matter their race, gender, or socio-economic status?
- Do you help build in each and every one of your students a positive mathematics identity and a high sense of agency?
- Do you emphasize and honor creative problem solving in your classroom alongside procedural fluency and conceptual understanding?
- Are you working to dismantle structural obstacles in your school or district that might be denying certain students access to upper-level mathematics courses?
If we continually ask ourselves these questions and act on them, then we increase the likelihood we will find, encourage, and support the “hidden figures” in each of our classrooms.
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Teaching Channel Videos ... and more
Grades K-2
Which one doesn't belong?
Grades 3-5
Creating community
Grades 6-8
For the mixed ability classroom
Grades 9-12
A classroom strategy
Grades 9-12
Work with transformations
All Grades
Improving math learning
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Which One Doesn't Belong?
A group of math teachers has created a variety of these visuals for use in a variety of grade levels. Mary Bourassa has created a Which One Doesn't Belong website which is continuously growing as contributors add more activities, and Christopher Danielson has just published a children's shape book and teacher's guide by this title.
These puzzles provide a fun way to get students involved with standards 1, 3, and 6 of the Standards for Mathematical Practice.
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Play as a Test Prep Tool
Standardized tests can be a wonderful teaching tool to enrich and deepen classroom learning.
What?! The prevailing wisdom is that standardized testing drains the life out of a classroom, saps students of interest and engagement, brings on unnecessary and at times crippling stress, and limits the view of what students are really learning in school.
Teaching to the test is a problem, for sure. But using the format of a standardized test as a teaching tool can enhance student learning—the question is how to do this in a way that captures students’ interest.
Here are a few ways to use the standardized test format to promote student engagement.
Play With Question Stems
Have students create the answer responses to a question stem, thinking carefully about “wrong” answers and finding the right language to construct the “correct” response. This is a highly analytical exercise and challenges students to really know and understand the concept being addressed in the question.
Flip the Question
Have students construct the question based on the answer responses. This forces students to identify the patterns and themes evident in the answer responses and thus arrive at the big idea in the question.
A No-Stakes Review
At the end of a class in a particular subject area, have students answer one multiple-choice, standardized-test type of question to see if they grasped an idea covered in class. This is a good way to garner immediate feedback. Time columnist Annie Murphy Paul shares the example of Columbia Middle School in St. Louis, Missouri, where teachers have students take a quick, no-stakes quiz—one that isn’t graded—at the end of each class to see what they learned.
The Quiz Show Format
Play Who Wants to Be a Millionaire? with multiple-choice questions. The popular ’90s TV show invited participants to answer a series of questions, sometimes enlisting the help of peers through the Call a Friend option, in which the participant could call a friendly source of information. The show also employed the 50/50 option, where two incorrect responses were eliminated from the answer list so that the participant could choose between just two options. Teachers can break the class into teams to play this game. In a more modern version, the Call a Friend option could give students one minute to Google the answer, forcing them to use intelligent search language to find the right answer. Or students could instead ask a friend for help.
Build Your Own Test
Give the class a mixed-up practice test, with the questions scrambled and in no apparent order of difficulty. Have teams of students reorder the questions, moving from easiest to hardest, being prepared to explain and defend why a certain question was easy or difficult. This also invites the students to consider the fact that on a standardized test all questions are equal, with no single question having more value than the others. Many students get hung up on hard questions and spend too much time on them instead of moving through the test to answer as many questions correctly as they possibly can.
Dispute the Question
Have students debate the merits of the wording of a particular question to find flaws, biases, or shortcomings and then rewrite the question with more careful wording.
Building experiences for students to play with a test can help to defuse anxiety, create familiarity and comfort, offer concrete strategies for success, promote collaboration and problem solving, and open up important conversations around taking standardized tests.
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Engaging Resources for Students from NCTM
Pre-K - 2
Thinking about numbers using frames of five can be a helpful way to learn basic number facts. The four games that can be played with this applet help to develop counting and addition skills.
Grades 3-5
A tessellation is a repeating pattern of polygons that covers a plane with no gaps or overlaps. What kind of tessellations can you make out of regular polygons?
Grades 3-8
Analyze data with bar graphs, line graphs, pie charts, and pictographs. Multiple rows and columns of data are allowed. Select which set(s) to display in a graph, and choose the type of representation.
Grades 6-8
A net is a two-dimensional figure that can be folded into a three-dimensional object. Which of the nets below will form a cube? Click on any net, and determine whether or not it can form a cube. An animation will provide further explanation.
Grades 6-12
Use tiles to represent variables and constants, learn how to represent and solve algebra problem. Solve equations, substitute in variable expressions, and expand and factor. Develop a better understanding of algebra.
Grades 9-12
Analyze data with box plots, bubble graphs, scatter plots, histograms, and stem-and-leaf plots. You can enter multiple rows and columns of data, select which set(s) to display in a graph, and choose the type of representation.
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Beat the Summer Math Slump
In a recent blog post, Leah Shafer shares suggestions from three Harvard faculty members for ways families can help reduce summer math loss: (1) highlight the math in everyday activities; (2) read short math stories together; (3) play math games -- including popular board games; and (4) find small ways to practice at home.
Summer math practice packets (and answer keys) have been prepared for students entering first grade all the way up to students entering Algebra 1. Making these practice packets available digitally reduces the time and costs associated with making paper copies, and students who take their Chromebooks home over the summer can use these devices to help keep their math skills sharp.
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Research Base
Alexander, K., Entwisle, D., and Olson, L. (2007). Lasting consequences of the summer learning gap. American Sociological Review, 72(2), 167-180.
Bourassa, M. (2013). Which one doesn’t belong? Retrieved from http://wodb.ca/index.html
Cooper, H., Nye, B., Charlton, K., Lindsay, J., & Greathouse, S. (1996). The effects of summer vacation on achievement test scores: A narrative and meta-analytic review. Review of Educational Research, 66(3), 227-268.
Danielson, C. [Teaching Channel]. (2016). Which one doesn’t belong? [Video file]. Retrieved from https://www.teachingchannel.org/videos/evidence-supported-claims-nsf
Digital Promise (Producer). (2017, March 1). Research@Work: Improving math learning with technology. [Video file]. Retrieved from https://www.youtube.com/watch?v=STSBBC4UEXM
Evans, P.E. [Teaching Channel]. (2012). Varied tasks for the mixed ability classroom. [Video file]. Retrieved from https://www.teachingchannel.org/videos/mixed-ability-classroom-management
Larson, M. (2017, March 21). Do you see and engage your hidden figures? [Blog post]. Retrieved from http://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Matt-Larson/Do-You-See-and-Engage-Your-Hidden-Figures_/
Levinson, M. (2017, March 8). Play as a test prep tool. [Blog post]. Retrieved from https://www.edutopia.org/blog/play-with-standardized-test-prep-matt-levinson
National Council of Teachers of Mathematics. (n.d.). NCTM Illuminations. Reston, VA: Author.
Shafer, L. (2016, June 24). Summer math loss: Why kids lose math knowledge, and how families can work to counteract it. [Blog post]. Retrieved from http://www.gse.harvard.edu/news/uk/16/06/summer-math-loss
Teaching Channel (Producer). (2015). Collaborative work with transformations. Retrieved from https://www.teachingchannel.org/videos/transformations-lesson-ccssmdc
Teaching Channel (Producer). (2014). Using extroverts. [Video file]. Retrieved from https://www.teachingchannel.org/videos/have-outgoing-students-lead-engagement-nea
Teaching Channel (Producer). (2016). Creating a community of learners. [Video file]. Retrieved from https://www.teachingchannel.org/videos/community-of-learners-cisco