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Blending Technology with Paper and Pencil

My geometry class is 1:1 this year; each student has her own MacBook Air. Students share responses to questions digitally in class using TI-Nspire Navigator for Networked Computers. Students explore mathematics using TI-Nspire dynamic graphs and geometry software. Students explore mathematics and share responses digitally using Demos Activity Builder. We use Canvas, an online learning management system, for assignments. We use Google Drive for sharing electronic documents with each other, and we use MathXL, online homework with built-in learning help, to practice mathematics. What place does pencil and paper have in my students’ learning and understanding of mathematics?

Even though many of the tasks that my students do for geometry take place digitally, I am convinced that pencil and paper plays an important role in how much mathematics my students not only learn but also remember. In a Wall Street Journal article, “Can Handwriting Make You Smarter?“, Robert Lee Hotz reports that students who take notes by hand usually outperform students who type notes when assessed more than one day after the class period. Students who type notes quickly type everything the professor says, but students who handwrite notes have to process the information while they are hearing it to select what is important to remember (Hotz 2016).

Hotz cites the work of Mueller and Oppenheimer published in Pyschological Science. Their research studies showed that students who took notes by hand performed better on conceptual questions than those who took notes on a laptop. Students performed about the same on factual questions. Their hypothesis for why is that students who take notes by hand choose which information is important to include in their notes, and so they are able to study “more efficiently” than those who are reviewing an entire typed lecture (Mueller and Oppenheimer 2014). Note: These studies are on college students; I have found little research on grade school students.

For several years now, my students and I have been learning how to learn mathematics using the Standards for Mathematical Practice. MP8, “look for and express regularity in repeated reasoning”, has pushed me to think about having students record what they see instead of just noticing and discussing it.

SMP8 #LL2LU Gough-Wilson

One of the ways that I’ve learned to talk about “look for and express regularity in repeated reasoning” is to ask students to notice what changes and what stays the same as we take a dynamic action on a geometric figure. Consider a recent learning episode from my classroom.

Students were told that our learning intention was “I can look for and express regularity in repeating reasoning”. The content was conceptual development of the equation of a circle in a coordinate plane using the Pythagorean Theorem. I did not share that specific content with students up front, however, because I wanted it to be revealed as the lesson progressed. I showed them a dynamic right triangle in the coordinate plane.

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What changes? What stays the same?

I could have let them simply discuss what they noticed. But instead I asked them to “Notice & Note”, using words, pictures, and numbers to write and sketch what they saw.

Then I asked them to share what they noticed with a partner and add to their own notes as desired.

 

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Our classroom discussion revealed that the equation of the circle formed by tracing point P was x2 + y2 = 52.

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Students continued to “Notice & Note” as they moved a circle around in the coordinate plane. What changes? What stays the same?

 

Notice & Note 3Notice & Note 4

As we moved P around in the coordinate plane, and then as they later moved the circle around in the coordinate plane students noted what they saw. Eventually, students generalized the center-radius form of an equation.

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Notice & Note, by Kylene Beers and Robert E. Probst, is a guide of signposts (strategies) for close reading of text. Students are taught signposts to notice while they are reading, and they are asked to stop reading and note what the signpost might imply. “Again & Again” is one signpost. Do you notice an event in the text that keeps happening again and again? Do you notice a phrase in the text that is repeated again and again? Stop reading, note it, and think about what that might mean (Beers and Probst 2013). How might we take advantage of the ways that students are learning to read text in their English Language Arts (ELA) classes to guide students in inquiry based exploration of mathematics?

In her online course, Sunni Brown, author of The Doodle Revolution, states that “Tracking content using imagery, color, word pictures and typography can change the way you understand information and also dramatically increase your level of knowledge and retention” (Brown 2016). How do we make tracking content using words, pictures, and numbers a reality in the 1:1 classroom? My experience is that it doesn’t happen without deliberate emphasis on its importance.

In Reading Nonfiction, Beers and Probst write “When students recognize that nonfiction ought to challenge us, ought to slow us down and make us think, then they’re more likely to become close readers” (Beers and Probst 2016). Our ELA counterparts are on to something. Effective classroom instruction is not just about creating learning episodes for our students to experience the mathematics using the Math Practices. Effective classroom instruction incorporates practices that will help students remember what they are learning longer than for the next test.

 

As I think about our district’s continued implementation of 1:1 technology, I am convinced that we need to pay attention to when we are asking, encouraging, and requiring students to use pencil and paper to create a record of what they are learning. I am interested in thinking more about how we might blend the use of dynamic graphs and geometry software with Notice & Note – using words, pictures, and numbers, along with color, so that students not only have a record of what they are learning but also have a better chance of remembering it later. And so, the journey continues …


References

Beers, G. Kylene, and Robert E. Probst. Notice & note: Strategies for close reading. Portsmouth: Heinemann, 2013. Print.

Beers, G. K., & Probst, R. E. (2016). Reading nonfiction: Notice & note stances, signposts, and strategies. Portsmouth: Heinemann.

Brown, S. (n.d.). Visual Note-Taking 101 / Personal Infodoodling™. Retrieved April 25, 2016, from http://sunnibrown.com/visualtraining

Hotz, Robert Lee. “Can handwriting make you smarter?” The Wall Street Journal. 04 Apr. 2016. Web. 25 Apr. 2016.

Mueller, P. A., and D. M. Oppenheimer. “The pen is mightier than the keyboard: Advantages of longhand over laptop note taking.” Psychological Science 25.6 (2014): 1159-168. Web.

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A Heuristic Approach to Angles in Circles

I am taking a qualitative research class right now, and my mind is full of lots of new-to-me words (many of which my spell checker doesn’t know, either): hermeneutics, phenomenology, ethnography, ethnomethodology, interpretivism, postpositivism, etc. One that has struck me is heuristic, the definition of which I can actually remember because I try to teach heuristically. (The word does not yet roll off of my tongue, but the definition, I get.)

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On Monday, our content was G-C.A Understand and apply theorems about circles

  1. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

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We started with a Quick Poll. I asked students for their best guess for the angle measure. I showed the results without displaying the correct answer, noting the lowest and highest guesses.

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Students moved to the technology. What happens to the angle measures as you move the points on the circle?

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They moved to the next page, which revealed more information. What happens to the angle measures as you move the points on the circle?

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I sent the poll again. There was one team who hadn’t answered yet, so I made a brief stop by their table. Last semester, I remember reading something about how a certain example might give students the eyes to see what you’re trying to get them to see. So we moved the points around to look something like this.

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If you have 49 and 43, how can you get 46?

Changing the numbers purposefully helped them see.

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I sent one more poll before we talked about why.

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So we gave our best guess, and then we used technology to explore. Students practiced MP8 I can look for and express regularity in repeated reasoning as they noticed what stayed the same and what changed with an angle whose vertex is in the center of the circle. They generalized the result. But we hadn’t yet discussed why that happens.

Students practice MP7 I can look for and make use of structure. By now they know our mantra for MP7: What can you make visible that isn’t yet pictured?

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I saw a line constructed parallel to the given line, which made alternate interior angles visible.

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I saw a chord drawn that made a triangle visible.

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I asked students to write down everything they knew about the angles in this diagram.

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They made suggestions about what we know. They didn’t say the relationships exactly like I would. I wrote them down anyway. They didn’t recognize the exterior angle of the triangle and so ending up proving the Exterior Angle Theorem again off to the side. I wrote it down anyway.

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And so the journey continues, always trying to enable my students to discover or learn something for themselves (and sometimes succeeding) …

 
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Posted by on February 9, 2017 in Circles, Geometry

 

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Hinge Questions: Dilations

Students noticed and noted.

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I wanted to be sure that they could answer a dilations question based on their observations. I had two questions premade in my set of Quick Polls. Which question would you ask?

In the past, I would have asked both questions without thinking.

I am learning, though, to think more about which questions I ask. If we only have time to ask a few questions, which questions are worth asking?

From slide 34 in Dylan Wiliam’s presentation at the SSAT 18th National Conference (2010) “Innovation that works: research-based strategies that raise achievement”.

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I decided to send the second poll. I decided that if they get that one right, they can both dilate a point about the origin and pay attention to whether they are given the image or pre-image. If I had sent the second poll, I wouldn’t know whether they could both do and undo a dilation.

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Next we looked at this question.

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Students worked on paper first.

Then some explored with technology.

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What do you want your students to know about the relationships in the diagram?

What question would you ask to see whether they did?

I asked this question to see what my students were thinking.

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And so the journey to write and ask and share and revise hinge questions continues …

 
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Posted by on December 20, 2016 in Coordinate Geometry, Dilations, Geometry

 

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Notice & Note: Dilations

How do you give your students the opportunity to practice MP8: I can look for and express regularity in repeated reasoning?

SMP8 #LL2LU Gough-Wilson

We started our dilations unit practicing MP8, noticing and noting.

 

Dilations 1.gif

What would you want students to notice and note?

How do students learn what is important to notice and note?

An important consideration when learning with self-explanation is to look at the quality of the explanation itself. What are the students saying or writing? Are they just regurgitating bits of text or making connections to underlying principles? Do the explanations contain predictions about what is going to happen, try to go beyond the given instruction or do they just superficially gloss over what is already there? Students who make principle-based, anticipative, or inference-containing explanations benefit the most from self-explaining. If students seem to be failing to make good explanations, one can try to give prompts with more assistance. In practice, this will likely take iteration by the instructor to figure out what combination of content, activity and prompt provides the most benefit to students. (Chiu & Chi, 2014, p. 99)

We had a brief discussion about what might be important to notice and note. We’ve also been working on predictions, thinking about what you expect to happen before trying it with technology:

What happens when the center of dilation is on the figure, outside the figure, and inside the figure?

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What happens when the scale factor is greater than 1? Equal to 1? Between 0 and 1? Less than 0?

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I observed, walking around the room and using Class Capture, selecting conversations for our whole class discussion.

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Here’s what NA noticed and noted.

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We looked at Hannah’s Rectangle, from NCSM’s Congruence and Similarity PD Module. Students had a straightedge and piece of tracing paper.

Which rectangles are similar to rectangle a? Explain the method you used to decide.Hannahs Rectangle.png

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What would you do next? Would you show the correct responses? Or not?

Would you start with an incorrect answer? or a correct answer?

Would you regroup students based on their responses?

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I started with a student who didn’t select G and then one who did. Then I asked a student who selected C to share why he chose C and didn’t choose F. We ended by watching Randy’s explanation on the module video.

And so the journey continues, always wondering what comes next (and sometimes wondering what should have come first) …


Chiu, J.L, & Chi, M.T.H. (2014). Supporting self-explanation in the classroom. In V. A. Benassi, C. E. Overson, & C. M. Hakala (Eds.). Applying science of learning in education: Infusing psychological science into the curriculum. Retrieved from the Society for the Teaching of Psychology web site: http://teachpsych.org/ebooks/asle2014/index.php

 

 
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Posted by on December 19, 2016 in Dilations, Geometry

 

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Introduction to Curve Sketching, Part 2

While students were working on this antiderivative from a Desmos Activity called Sketchy Derivatives, I heard several students ask how they know where to place the y-intercept.

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We didn’t have time to answer that question during the first lesson of the unit, but we started with it during the second lesson.

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Should the cubic start decreasing or increasing? How do you know?

[Yes, I do know that I can anonymize the names. However, by the time I thought of that, these two responses were no longer adjacent. It would be nice to be able to drag the responses to different locations in case you want to compare/contrast several specific responses at the same time. ☺]

Then we looked at these.

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They all have the same basic shape. Is one more right than the other?

Students began to think about all of the curves that have y’=2x, and so while they know something about the importance of the constant, one won’t be more right than the other until we learn about area under the curve.

I had the Second Derivative Grapher from Calculus Nspired queued during the previous class, but I decided it would be better for students to “do” instead of “discuss” for the last few minutes of class. We looked at it next to make the relationship between the second derivative and concavity of the original function more clear.

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Students put all of this information together to begin to analyze a function given the graph of its derivative. The results of our formative assessments seem to indicate that students have a better understanding of the relationship between f-f’-f” than they have in the past. I’ll know for sure later today, though, after their summative assessment. And so the journey continues …

 
 

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Introduction to Curve Sketching, Part 1

Learning Intentions:

Level 4:

I can use the graph of the derivative to sketch a graph of the original function.

Level 3:

I can use the graph of the original function to deduce information about the first and second derivatives.

I can use the graph of the derivative to deduce information about the second derivative and the original function.

I can use the graph of the second derivative to deduce information about the first derivative and the original function.

Level 2:

I can determine when a function is concave up or concave down and where it has points of inflection.

Level 1:

I can determine when a function is increasing or decreasing and where it has maxima and minima.

We were on the first day of a new unit. I included two questions on the opener to ensure students know what we mean by increasing/decreasing and concave up/concave down intervals. As expected students were familiar with increasing/decreasing and not so familiar with concave up/concave down.

Based on the results, we discussed what it means to be concave up and concave down. Someone asked how we would be able to tell for sure where the graph changes concavity, which we get to learn during the unit.

We started the lesson with a few Quick Polls for students to determine which graph was the derivative, given the graphs of a function and its derivative. The polls were based on Graphical Derivatives from Calculus Nspired. I sent the poll, asked students to answer individually, stopped the poll, asked students to explain their thinking to a partner. If needed, I sent the poll again to see whether they wanted to change their response after talking with their partner. I had 6 polls prepared. I sent 3.

I listened while students shared their thinking. I selected three conversations for the whole class.

  1. A student who knew which was which based on the power rule, which she learned during the last unit.
  2. A student who knew that the slope of the tangent line at the minimum of the parabola should be zero, which is the value of the line at z=0.
  3. A student who noticed that the line (derivative) was negative (below the x-axis) when the parabola was decreasing and positive (above the x-axis) when the parabola was increasing.

Again, as I listened to the pairs talking, I selected a few students to share their thinking with the whole class.

  1. The first student who shared used the maximum and minima to determine which had to be the derivative, since the derivative is zero at those x-values.
  2. The second student thought about what the slope of the tangent line would be at certain x-values and whether the y-values of the other function complied.
  3. A third student volunteered a fourth student to discuss her thinking: she noted that the graph of the function (b) changed concavity at the max/min of the derivative (b).

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After students talked, I sent the poll again to see if anyone was convinced otherwise.

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Two students briefly discussed how they used increasing/decreasing and concavity to determine the derivative.

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Next we began to solidify what increasing/decreasing and concave up/concave down intervals look like using Derivative Analysis from Calculus Nspired.

I asked students to notice and note.

Where is the function increasing? Where is it decreasing?
What is the relationship between the slope of the tangent line and where the function is increasing and decreasing?

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Where is the function concave up? Where is it concave down?

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What does the tangent line have to do with where the function is concave up and concave down?

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Can you look at a graph and estimate intervals of concavity?

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I was able to see what students were noting on paper and hear what they were noting in our conversation, but I didn’t send any polls during this part of the lesson.

Next we looked at Derivative Grapher from Calculus Nspired.

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We changed the graph to f(x)=cos(x). We already know the derivative is f’(x)=sin(x). What if we were only given the graph of the derivative? How could we use that graph to determine information about the original function?

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I had more for us to discuss as a whole class, but I wanted to know what they had learned before the class ended. I used a Desmos Activity called Sketchy Derivatives to see what students had learned – given a function, sketch its derivative; and given a derivative, sketch an antiderivative. The original activity was from Michael Fenton. I modified it to go back and forth between sketching the derivative and antiderivative instead of doing all derivatives first and all antiderivatives second, and I added a few questions so that students could begin to clarify their thinking using words.

We spent the last minutes of class looking at an overlay of some of their sketches.

Could you figure out exactly where to sketch the horizontal line?

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Most students have the vertex of the parabola near the right x-coordinate. Should the antiderivative be concave up or concave down?

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Most students have the derivative crossing the x-axis near the correct location.

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The bell rang. Another #lessonclose failure. But thankfully, there are do-overs as the journey continues …

 

 
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Posted by on October 17, 2016 in Applications of Differentiation, Calculus

 

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MP5 – The Center of the Circle

How do you give students the opportunity to practice “I can use appropriate tools strategically”?

MP5

How would your students find the center of a circle?

Every year, I am amazed at the connections students make between properties of circles that we have explored and what the center of the circle has to do with those properties.

We started on paper.

Some students moved their thoughts to technology.

Whose work would you select for an individual and/or whole class discussion?

Could we use the tangents to a circle from a point to find the center of the circle?

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Could we use the intersection of the angle bisectors of an equilateral triangle inscribed in a circle to find the center of the circle?

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Could we use the perpendicular bisector of a chord of a circle to find the center of the circle?

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Could we use the intersection of the perpendicular bisectors of a pentagon circumscribed about a circle to find the center of the circle?

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Could we use the intersection of the perpendicular bisectors of several chords of a circle to find the center of the circle?

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Could we use a right triangle inscribed in a circle to find the center of the circle?

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And so the journey continues …

 
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Posted by on September 15, 2016 in Circles, Geometry

 

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