“Technology and Noticing” – Psychology of Mathematics Education – North American Chapter

PMENA 2015 - Noticing and Technology - Final

I had a great time at PMENA-37 this past weekend! What a supportive, engaging, and productive space for learning about mathematics education – rather, immersing oneself into the mathematics education community.

Friday afternoon, I presented my poster, “Digital resources in mathematics: Teachersconceptions and noticing” and I learned a lot through conversations I had with visitors to my poster. You can read the overview of this poster in PMENA-37 Proceedings (p. 1256).

This poster represents a moment in time in my dissertation study. I am beginning to look at what teachers notice (or paid attention to, or found important) in online tools and resources by first considering how they responded in our pre- and post-evaluations in the online master’s-level course “Learning Mathematics with Technology.” We (the instructor and I, as a teaching assistant) gave students a broad prompt:

Assume you are considering using these tech tools in your teaching. Evaluate the three tools to decide whether you would recommend using them. Write an evaluation of the three tech tools for an audience of other teachers who might be considering using them.  Make your recommendation clear: which (if any) would you recommend? Argue your position. You will need to decide which elements, features, or characteristics of each tech tool to use in supporting your argument.

I considered the teachers in my study who chose the addition set of tools, and as I read and re-read their evaluations I created a list of tool features that the teachers paid attention to: Purpose of tool; Classroom use; Understanding (e.g., instrumental, relational); Thinking (e.g., critical, level); Engaging / Aesthetically pleasing / Distractions; Directions and Instructions; Customizable; Progress / Tracking; Preparation; Age / Grade-level; Interaction / Manipulative; Feedback / Responsiveness; Ease of Use / Accessibility; Differentiation / Learning styles / Learning needs; CCSSM / Standards Alignment; Mathematical representation; Teacher Support / Resources; Breadth / Variation. These were too many for the poster, so I narrowed them down to 12, and sorted them according to whether they seemed to be general, pedagogical, or mathematical characteristics (but recognizing the overlap in the categories, similar to the TPACK framework):

  • General: Customizable; Engaging / Aesthetically pleasing; Ease of Use / Accessibility
  • Pedagogical: Differentiation / Learning styles/needs; Progress / Tracking; Feedback / Responsiveness; Teacher Support / Resources
  • Mathematical: Mathematical Purpose; Standards Alignment; Understanding / Thinking; Interaction / Manipulative; Mathematical representation

In the tables, I noted whether teachers mentioned a particular characteristic was included or favorable included in the tool (Y), was not included or included in an unfavorable way (N), and whether the teacher did not mention a characteristic (–). Because of the limited space of the poster, I chose three mathematical features to show teacher comments about: Mathematical purpose, Understanding / Thinking, and Interaction / Manipulative. I included comments for each tool from each teacher in the pre-evaluation (left) and the post-evaluation (right).

Discussion of Results (which are shown in the poster above)

It goes without saying that the online tool itself seemed to impact the features that teachers paid attention to and I had chosen the three tools to be different in (hopefully) interesting ways. One tool is built around a ten-frame representation of addition, and includes a symbolic representation (horizontal addition statements). One tool is a quizzing software that includes a symbolic representation (horizontal statement) of addition only. One tool is built around a base-10 blocks representation of addition, and includes a symbolic representation (vertical addition statement). The tools differ in a number of ways and the teachers noticed many of their differences. There is evidence that teachers moved, not just to noticing more characteristics, but also to more profound noticing by the end of the course.

One interesting result was the teachers’ attention to thinking and understanding in the Math Trainer applet. In the pre-evaluation, none of the four teachers commented on opportunities for students to think or develop understanding. In the post-evaluation, all four teachers commented on Math Trainer’s support of instrumental understanding (we read Skemp (1976) early in the course), and on the lack of opportunities to “explore or gain a deeper understanding,” or for “critical thinking.”

Another interesting result is the change in Nicole’s use of words from the pre-evaluation to post-evaluation in her description of the mathematical purpose of each tool. She did not change words at all in her description of Math Trainer’s purpose “to practice math facts,” but changed from practice to explore for NCTM Illuminations: Ten Frame Addition and NLVM: Base Blocks Addition. Her description of the purpose of NCTM Illuminations: Ten Frame Addition in the pre-evaluation was “to practice using a 10-frame to work with numbers” and in the post-evaluation was “to work with a ten-frame (or five-frame option) to explore addition and subtraction of whole numbers.” For NLVM: Base Blocks Addition, she wrote “to practice addition with base-ten blocks” in her pre-evaluation and “to explore addition with base-ten blocks” in her post-evaluation.

PMENA 2015 – Noticing and Technology – Final (pdf of poster)

PMENA 2015 Proceedings – Noticing and Technology (pdf of proceedings paper)


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.

Creating a Number Line in PowerPoint

Making a nice number line in PowerPoint is quick and easy – it should only take about 10-15 minutes. These instructions are really just a frame for learning a few neat formatting tricks that will make using PowerPoint for graphical interfaces like posters (or even presentations!) a snap.The first thing is to “Insert” a “Shape” (in this case a line) and choose the color, thickness, and other effects using “Format” in “Drawing Tools”. To make sure the line is straight, hold the “Shift” key down as you drag it out.PowerPoint-NumberLine-01Make a smaller line for the tick marks, and copy paste however many times you’d like. The position doesn’t matter – we’ll straighten those out in a minute.PowerPoint-NumberLine-02Left-click and drag the mouse  to highlight all of the ticks fully. When you release the mouse button, all of the ticks will be selected. PowerPoint will only select shapes (or images) if it is entirely within the highlighted area, so you can be very selective by leaving out even a tiny particle of a shape.PowerPoint-NumberLine-03If you select objects that you didn’t want to select, just hold the <CTRL> key down and click on them to deselect.PowerPoint-NumberLine-04Once you’ve selected everything, go to the “Format” window and look to the far right for the “Align” menu.  This menu gives you many options and, if you haven’t yet,  you should definitely experiment and figure out what they all do and think about how they can be useful.PowerPoint-NumberLine-05These tickmarks need to be aligned horizontally, so choose “Align Middle” (although really top or bottom would also work fine in this case.)PowerPoint-NumberLine-06Now, click anywhere else on the slide to deselect all of the ticks. Choose one and drag it all the way to one side so that the tick on the left and the tick on the right are in the desired positions.  (You can drag it around without ruining alignment by holding the <Shift> button down.PowerPoint-NumberLine-07Once they are in place, go back to the “Align” menu and choose “Distribute Horizontally”.  Congratulations! You have a number line (albeit blank).PowerPoint-NumberLine-08Adding the numbers is easy too – just use text boxes and the align menu to get them straight and in place.

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.com.

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.

Math (and more) with Wolfram|Alpha

Numbers and Operations

Wolfram|Alpha can of course be used for simple calculations but it will go far beyond that, trying to give everything to you that you could be looking for.



Here, a relatively simple calculation results in the numeric answer (7), its verbal representation (seven), as well as a manipulatives representation and number line representation of the addition process. It also includes speed for computation at different ages which is fun to think about.


Wolfram|Alpha will attempt to guess what the user wants but will tell you what it’s assuming. For example, in this case it said “Assuming trigonometric arguments in degrees” and offered to use radians instead.  It also provides links to “Related Queries” on bottom of the results page.


If we up the ante a bit, and try a quadratic equation:


It gives us the type of geometric figure (parabola) as well as a graph:


You can see that on the “geometric figure” field there is a button for “Properties”. Pressing this will yield additional information including the focus, vertex, semi-axis length, focal parameter, eccentricity, and directrix.


The graphs each have buttons to “enable interactivity” (although apparently that is now a WolframAlpha Pro feature).

But wait! That’s not all…

It also shows an alternate form, derivatives and the global minimum.


Typing different forms will result in different attempts to answer the user’s question.  Try entering just “x^2 – 2” or “solve x^2 – 2 = 0”.

Notice that for solutions, it will offer the option of “Show Steps” which will walk you step by step through the solution process (although it looks like this feature is also now part of Wolfram|Alpha Pro).

An advantage is that many examples can be shown quickly (or explored), so students might make conjectures and look for patterns, as they experiment with values of parameters.

Data and Analysis

For example, you can search your name and see how popular it was when you were born.  I found that my parents were almost 15 years ahead of the times – very few girls my age are named Eryn.


It also states that, with my name, it is most likely that I was born in 2000.  Thus, my name will be helpful when I start pretending to be 25 years younger than I really am someday.

You can also compare two names (or really two of anything!) by typing “compare ___ and ____” or just “___ | ____”.


For example, I can type “compare provo and east lansing” or just “provo | east lansing”.  Or I can ask to compare a specific characteristic like population or weather:

Wolfram|Alpha uses the most up-to-date data that is available online. In this case, it tells the user that Provo’s weather was last updated 31 minutes ago and East Lansing’s was last updated 33 minutes ago.


To sum up, Wolfram|Alpha can be useful in a number of ways and can add relevance to mathematics application questions.  It can also provide some useful information to students, but care should be taken to ensure that they are thinking about the information provided rather than accepting it at face value.

To find more information or participate in an online mathematics educators community centered around teaching math with Wolfram|Alpha, visit Wolfram|Alpha for Educators (some lesson plans) or the Wolfram Demonstrations Project (interactive activities) or even Wolfram MathWorld (definitions).

Sine Waves for Musical Scales

While I was teaching trigonometry, I discovered Wolfram Demonstration Project and determined to write a demonstration for my trigonometry course.

The idea behind demonstrations is that they have the power of Mathematica but a student can interact with the demonstration without installing Mathematica, a software that is expensive and complex to use.

I wrote a demonstration for students to interact with a trigonometric function that modeled piano tones (kind of):

Access my demonstration: http://demonstrations.wolfram.com/SineWavesForMusicalScales/

The most common tuning system in Western music is the twelve-tone equal temperament scale. In this example, notation is used assuming octaves on a standard piano keyboard with 88 keys, numbered from the left end of the keyboard. Then middle C, for example, is the 40th key. The 49th key, A, is the reference pitch with a frequency of 440 Hz. The frequencies of the other keys in the octave come from this formula: , where is the frequency of the key. We can model the tones using a sinusoidal wave, , with .