Modeling 5 Practices – Monitoring

Attachment: Measurement Applets – Monitoring Sheet

In Spring 2015, I taught one section of a mathematics methods course for senior preservice K-8 teachers (PSTs). The seniors spent about 5 hours each week in a classroom – most were in the same classroom all year. In two large assignments, they created high-level tasks, wrote lesson plans, and then implemented their lesson plans. Throughout the process they used, as resources: Van de Walle (2013), Smith, Bill, and Hughes (2008), and Stein, Engle, Smith, and Hughes (2008). One requirement of the lesson plan was that it explained how they would incorporate the “5 practices” to lead a productive mathematics discussion.

My seniors struggled with anticipating and monitoring, so I developed an in-class task where they could experience the monitoring, selecting, sequencing, and connecting practices. I did the “anticipating” and created monitoring sheets. The seniors engaged in five tasks in pairs, using applets from (links are found in the attached monitoring sheets) on the classroom computers. They were in five groups, so for each task one group monitored their classmates and four groups engaged in the task.

After the five tasks, the seniors had ten minutes to talk in groups about how they would select and sequence the strategies they had seen to support a particular measurement learning goal. Then each group led a (unfortunately too-brief) five-minute discussion to support their classmates in making connections.

The applets were designed by the Strengthening Tomorrow’s Education in Measurement research project to support K-8 students in confronting misconceptions about particular aspects of measurement. I helped design the applets, and (for most of them) wrote the code to create the applets. My seniors thus were able to talk deeply about measurement strategies as well as to experience the decision-making process around the 5 practices.


Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a Lesson: Successfully Implementing High-Level Tasks. Mathematics Teaching in the Middle School, 14(3), 132–138.
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.
Van de Walle, J. A., Karp, K. S., Lovin, L. H., & Bay-Williams, J. M. (2013). Teaching student-centered mathematics: Developmentally appropriate instruction for grades Prek–2 (Vol. 1). Boston, MA: Pearson.
Van de Walle, J. A., Karp, K. S., Lovin, L. H., & Bay-Williams, J. M. (2013). Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Vol. 2). Boston, MA: Pearson.

Creating Technologies to Support Measurement

I worked with a student teacher who had been an undergraduate research assistant for the Strengthening Tomorrow’s Education in Measurement project, and was a recent graduate of Michigan State’s Teacher Education program. She is currently interning in a nearby classroom. She agreed, along with her mentor teacher, to design and implement a lesson with my support. Two redesigned applets resulted, and a third is possible. The link below connects to her lesson.

The student teacher had taught a lesson that incorporated van de Walle’s Crooked Path and Broken Ruler activities two weeks before the technology lesson (Van de Walle, 2013). She created crooked paths in her classroom using masking tape on the floor and on some tables. Students measured them, moving from path to path as they completed each measurement.

The student teacher felt that the STEM Jagged Path applets (see link above) would support the learning goals of her previous lesson by encouraging students to practice skills they had developed as well as deepening their earlier thinking. She asked all students to explore Activity 1, using Activity 3 as an extension for students who finished quickly.

The student teacher and I talked above another applet that could support students’ thinking. She said that her mentor teacher worked hard to encourage her students to confront misconceptions and change their thinking. We designed and created a third applet (shown in Activity 2, using the site link above) that would show the student two “wrong” ways to measure. We asked the students to explain what they thought was the thinking behind each method, and to explain which method they thought was correct – or if neither was correct, to explain what a correct method would be.

Supporting spatial measurement tasks with technology

Session presented at the Michigan Council of Teachers of Mathematics annual conference in Traverse City, MI.

Download slides here.

In this hands-on session, we introduced several sample tasks that included use of computer applets to strengthen understanding of length and area concepts. Many students struggle with length and area, using only pen and paper or physical manipulatives. In this session, we provided materials to support their learning (that will be available for use during and after the session).

We asked participants to use their devices to work through one sample lesson (Area of Rectangular Regions) including Launch / Explore / Summarize components. We gave participants two sample lessons that they could in their classrooms, along with applets designed and created by Strengthening Tomorrow’s Education in Measurement project.

Stehr, E.M., Gönülateş, F., & Siebers, K. (2013, August) Supporting spatial measurement tasks with technology. Paper presented at the Michigan Council of Teachers of Mathematics Conference, Traverse City, MI.

GeoGebra – Malfatti’s Problem

In Axiomatic Geometry, I and a partner explored Malfatti’s problem (see references below). I created a GeoGebra worksheet to support the presentation we later gave. I share some details here.


Malfatti originally asked this question is 1803 and the question, “How to arrange in a given triangle three non-overlapping circles of greatest area?” remained open until 1994.


As far back as 1384, in a manuscript of Italian mathematician Gilio de Cecco da Montepulciano a problem is presented of how to construct 3 circles, inside a triangle, that are tangent to each other and tangent to 2 edges of triangle. The same problem was posed by 18th century Japanese mathematician Ajima Naonubu. In 1803, Malfatti refined the problem to include minimizing the amount of area in the triangle “wasted”, i.e., constructing the largest circles possible. In 1826, Steiner (that sly old fox) developed a construction of three circles, tangent to each other and touching 2 edges of the triangle.  This was a solution to the problems posed by Montepulciano and Naonubu, but not Malfatti.  Stiener demonstrated 3 circles in a triangle but did not minimize the area “wasted”. In 1930, Lob and Richmond showed Malfatti’s circles do not produce the largest circles (i.e., minimal waste).  Optimal circles are obtained using greedy algorithm. In 1967, Goldberg showed that the Mafatti circles will never produce greatest area and the Lob and Richmond ‘greedy algorithm’ always produces the largest total area.  In 1994, Zalgaller and Los classified all the different ways 3 circles can be arranged in a triangle and developed a formula for determining which method of arranging the circles is optimal for a given triangle.  They also proved that there are 14 combinations of circles in a triangle that will yield optimal area.


Essentially, if there is room for a circle to ‘grow’ then the system is NOT rigid.  The two examples on the left are rigid, the one on the right is not, the middle circle could change radius or ‘grow’.


Zalgaller and Los proved the result systematically, reducing all possibilities to 14 combinatorially, non-identical, rigid configurations of three circles in a triangle. See cases below.


Malfatti’s Marble Problem asks:

  • How can we arrange in a given triangle three non-overlapping circles of greatest area?
  • For an Isosceles triangle, the first two circles are drawn as K1 and K2. Then we choose K3a or K3b depending on the relationship between sin(alpha) and tan(beta/2).
  • Play around with the triangle and tell me –
    • Do we use K3a or K3b when sin(alpha) > tan(beta/2)?
    • Do we use K3a or K3b when sin(alpha) < tan(beta/2)? W
    • hat do we do when sin(alpha) > tan(beta/2)?
    • Also, are the three circles always symmetric for an Isosceles triangle?



Andreatta, M, Bezdek, A, & Boronski, J (2010) The Problem of Malfatti: Two Centuries of Debate. The Mathematical Intelligencer.

Coolidge, J. L. (1923). Some Unsolved Problems in Solid Geometry. The American Mathematical Monthly, 30(4), 174-180.

Goldberg, M. (1967). On the original Malfatti problem. Mathematics Magazine, 40(5), 241-247.

Zalgaller, V. A., & Los’, G. A. (1994). The Solution Of Malfatti’s Problem. Journal Of Mathematical Sciences, 72(4), 3163-3177.

Presentation Reference

Humes, W.R. & Stehr, E.M. (2013). Malfatti’s Marble Problem. Presentation prepared for final project in D.S. Durosoy’s MTH 432: Axiomatic Geometry course.

Algebra and Graphing Calculators

Calculators are widely used in classrooms today to support student learning of mathematics.  Ten years ago, Dion et al. (2001) conducted a wide-spread survey of school calculator use.  Of the 4,568 schools that responded, 95% answered that they either require (46%) or at least allow (49%) the use of graphing or scientific calculators.

Both the Common Core State Standards in Mathematics (CCSS-M) and the National Council of Teachers of Mathematics (NCTM) Standards make strong recommendations for the use of technology in the classroom. Technology should be a tool “in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning” (NCTM, 2000, p. 24). The Common Core State Standards (2010) Standards for Mathematical Practice states, as an example of student tool use, that “mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator.  They detect possible errors by strategically using estimation and other mathematical knowledge” (p. 7).

The widespread use of calculators in the classroom is not necessarily equivalent to widespread effective use.  Milou (1999) surveyed 146 teachers and concluded that “successful integration of the graphing calculator into the mathematics classroom cannot take place without the aid of enthusiastic teachers.”

Both the NCTM and CCSS-M standards previously quoted encourage the use of technology as tools, when appropriate.  Hiebert et al. (1997)7) stated: ‘Students must construct meaning for all tools…. As you use a tool, you get to know the tool better and you use the tool more effectively to help you know about other things.’ (p. 54). This suggests that teachers and students both must learn to use technology such as graphing calculators by co-constructing that knowledge.

Calculators as Tools not “Dispensers of Answers”

In 1981 a study was conducted to study the process of student thinking in estimation tasks.  As one part of the study, the researchers asked 33 participants of various ages from child to adult, who had been identified as the top 10% of estimators in previous interviews, to estimate a calculation and then to use a calculator to check the answer.  They used a specially programmed HP-65 calculator that would make increasingly large errors in calculation.  In the study, 36% of females and 77% of males said something about the calculator being in error before they reached the last problem. One conclusion drawn by the researchers was that: “[t]he unwillingness of these good estimators to reject unreasonable answers suggests that a challenging task lies ahead in preparing students to be alert to unreasonable answers” (Reys, 1980, p. 231).

This study was redone by Glasgow and Reys (1998) almost 20 years later with 25 undergraduate participants each of whom had been identified as a top student in his or mathematics course.  In this second experiment, only 28% of the participants questioned the calculator before the last question.  Several of the participants in this second study were pre-service teachers.  One statement by the authors in the conclusion mentioned the importance of convincing these pre-service teachers “to question or reflect on their own mathematical thinking”.  The authors also argued that students needed to see calculators “as tools, not ‘dispensers of answers’” (p. 388).

In order to be able to use the calculator effectively in the classroom to support student learning of effective use of this tool rather than dependence on it, teachers must first construct their own knowledge of the calculator as a tool rather than as a sort of magic or “black box” that is unconnected to reality and is unpredictable.

Knowledge for Teachers

Chamblee, Slough, and Wunsch (2010) conducted a study to measure the concerns of in-service teachers before and after a year-long professional development program.  This program intended to support mathematics and science teachers in their use of graphing calculators, probes and sensors in the classroom.  90 teachers received 105 days of professional development throughout the school year and following summer.  The teachers also received a classroom set of graphing calculators, probes and sensors. The researchers tested 22 teachers before and after 105 days of professional development.  One of their findings was teachers needed professional development activities that allow them to learn how to use the graphing calculator in their own day-to-day classroom, instead of watching the professional development facilitator “show-and-tell” possible uses and activities.

This seems to suggest that seeing the calculator and its functionality is not enough to incorporate it into classroom life. The teacher needs opportunities to meaningfully construct her own knowledge – she needs to be involved in her own learning process in order to construct the knowledge necessary to strategically use the calculator in the classroom.

Support for Student Learning

In Doerr and Zangor (2000) this co-construction of the graphing calculator as a tool was studied in two pre-calculus classes all taught by the same teacher. The students used graphing calculators in class, along with probes and sensors for data collection.  The researchers suggested that both teachers and students can use a graphing calculator in different ways: as a computational tool, transformational tool, data collection and analysis tool, visualizing tool, or checking tool (p. 151).  As a computational tool, it is merely used to evaluate numerical expressions.  The danger is that students will see it only as a computational tool (Herman, 2007; Ruthven, 1995).  As a transformation tool, the calculator transforms “tedious computational tasks… into interpretative tasks” (p. 153).  In support of this role, the teacher must be often re-focusing the students’ attention away from the computation and to interpretation.  As a checking tool, the students might use the calculator to verify results acquired through different methods.  One danger of allowing students to use the calculator too often as a checking tool might be that it would lead to the students abdicating responsibility to their calculators, as discussed above in Reys (1980) and Glasgow and Reys (1998).

Pugalee (2001) also suggested that graphing calculators are not a magic device that result in student learning.  He argued that technology must be used with intentional discourse in order to be effective.  He performed a study using constructivism and graphing calculators in an algebra course for “at-risk” students.  The study covered only one section of 16 students, but the results were favorable. The use of graphing calculators, along with questioning, supported the students’ construction of mathematical knowledge. These results support Doerr and Zangor (2000) above since they also argued that the teacher needs to focus the students’ attention on interpreting calculator results instead of only focusing on the calculation in order to increase their mathematical knowledge.


CCSS-I. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Chamblee, G. E., Slough, S. W., & Wunsch, G. (2010). Measuring High School Mathematics Teachers’ Concerns About Graphing Calculators and Change: A Year Long Study. Journal of Computers in Mathematics and Science Teaching, 27(2), 183.

Dion, G., Harvey, A., Jackson, C., Klag, P., Liu, J., & Wright, C. (2001). A survey of calculator usage in high schools. School Science and Mathematics, 101(8), 427-438.

Doerr, H. M., & Zangor, R. (2000). Creating meaning for and with the graphing calculator. Educational Studies in Mathematics, 41(2), 143-163.

Glasgow, B., & Reys, B. J. (1998). The authority of the calculator in the minds of college students. School Science and Mathematics, 98(7), 383-388.

Harvey, J. G., Waits, B. K., & Demana, F. D. (1995). The influence of technology on the teaching and learning of algebra. The Journal of Mathematical Behavior, 14(1), 75-109.

Herman, M. (2007). What Students Choose to Do and Have to Say About Use of Multiple Representations in College Algebra. The Journal of Computers in Mathematics and Science Teaching, 26(1), 27-54.

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Wearne, D., Murray, H., . . . Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.

Hollar, J. C., & Norwood, K. (1999). The effects of a graphing-approach intermediate algebra curriculum on students’ understanding of function. Journal for Research in Mathematics Education, 30(2), 220-220.

Mayes, R. L. (1995). The application of a computer algebra system as a tool in college algebra. School Science and Mathematics, 95(2), 61-68.

Milou, E. (1999). The Graphing Calculator: A Survey of Classroom Usage. School Science and Mathematics, 99(3).

NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc.

O’Callaghan, B. R. (1998). Computer-intensive algebra and students’ conceptual knowledge of functions. Journal for Research in Mathematics Education, 21-40.

Pugalee, D. K. (2001). Algebra for all: The role of technology and constructivism in an algebra course for at-risk students. Preventing School Failure, 45(4), 171-176.

Pullano, F. (2005). Using Probeware to Improve Students’ Graph Interpretation Abilities. School Science and Mathematics, 105(7), 373-376.

Reys, R. E. (1980). Identification and Characterization of Computational Estimation Processes Used by Inschool Pupils and Out-of-School Adults. Final Report.

Ruthven, K. (1995). Pupil’s View of Number Work and Calculators. Educational Research, 37, 229-237.

Microsoft Mathematics and OneNote

(Note: Microsoft Mathematics can also be used independently of OneNote)

Microsoft OneNote has many excellent features that support note-taking and searchability, but there are too many features to list in one post. This post will focus on Microsoft Math and OneNote. First of all, there is a built in calculator in OneNote that can be used when typing.  Just type the calculations, add an equals sign and press enter:


(Note that trigonometric functions in OneNote are in degrees rather than radians.)  For a list of mathematical operations and supported functions, check out the informational page at

Microsoft OneNote makes it easy to ink mathematics using the “Ink to Math” option:

MSMath-02It does pretty well at guessing what you mean – but the more information it has, the  better it can guess.  So expect to see it adjust frequently as you write.


Once you are comfortable with the Ink to Math feature, using Microsoft Mathematics can give many more options.  Microsoft Mathematics is a free download and can be used independently, but once you have installed it then you can also install the add-in for OneNote. This gives many options.  You can use the add-in to graph equations.


You can also use it to factor or expand polynomial expressions, find derivatives or integrals, or even plot graphs in 2D or 3D:


The best part about this is that you can use this in class as a substitute for chalk and blackboard, and then print the notes to pdf so the students can use them as a resource, or to create their own activities in OneNote.