Notice and Note: Dilations

Are you familiar with Notice & Note: Strategies for Close Reading? Here’s a link to Heinemann’s Notice & Note learning community, and here’s a sample PDF. I wonder whether our Standards for Mathematical Practice are similar to the Notice and Note literary signposts.

It’s not enough to just read a text. We want students to read for understanding and comprehension. The literary signposts help students with close reading of a literary text.

Similarly, it’s not enough to just explore math with dynamic graphs and geometry. We want students to explore for understanding and comprehension. The math practices help students learn how to interact with a mathematical problem or concept … and what to notice.


Last week, we explored dilations.

What do you need for a dilation?

A figure, a point (which we’ll call the center of dilation), and a number (which we’ll call the scale factor)

We used our dynamic geometry software to perform a dilation.

1 11-15-2015 Image001

About what things might you be curious as you explore dilations?

(I thought of Kristin when I used the word curious.)

What happens when the center of dilation is inside the pre-image?

What happens when the center of dilation is on the pre-image? (on a side, on a vertex)

What happens when the scale factor is between 0 and 1?
What happens when the scale factor is negative?

How do the corresponding side lengths in the pre-image and image relate to each other?


I asked students to practice look for and express regularity in repeated reasoning as they explored the dilation. Do you know what it means to look for and express regularity in repeated reasoning?

Find a pattern.

Yes. Figure out what changes and what stays the same as you take a dynamic action on the dilation. Begin to make some generalizations about what you notice.

SMP8 #LL2LU Gough-Wilson.png

And don’t just notice, but actually note what you’re thinking.

The room got quiet as students noticed and noted their observations about dilations. I monitored student work both using Class Capture and walking around to see what students were noting.

(I promise I’ve tried to make it clear to students that dilation has 3 syllables and not 4 … but we do live in the South.)

Eventually, they shared some of their findings with their team, and then I selected a few to note their observations for the whole class.

BB showed us what happened when he perfomed a dilation with a scale factor of -1. He had noted that it was the same as rotating the pre-image 180˚ about the center of dilation.

SA talked with us about when the dilation would be a reduction. She had decided it wasn’t enough to say a scale factor less than 1 or a fractional scale factor but that we needed to say a scale factor between 0 and 1 or between -1 and 0.

FK showed us that when she drew a line connecting a pre-image point and its image, the line also contained the center of dilation.

PS noted that when the scale factor was 2, the length of the segment from the center of dilation to a pre-image point equaled the length of the segment from the pre-image point to its image.

When the scale factor was 3, the length of the segment from the center of dilation to a pre-image point equaled one-half the length of the segment from the pre-image point to its image.


We next determined a dilation and set of rigid motions would show that the two figures are similar.

20 Screenshot 2015-11-11 09.34.31

Translate ∆DET using vector EY.

21 Screen Shot 2015-11-17 at 5.25.08 AM

Rotate ∆D’E’T’ about Y using angle D’YA.

22 Screen Shot 2015-11-17 at 5.26.55 AM

Dilate ∆D’’E’’T’’ about Y using scale factor AY/D’’Y.

23 Screen Shot 2015-11-17 at 5.30.35 AM

Then we looked at dilations in the coordinate plane. I knew that my students had some experience with this from middle school, and so I sent a Quick Poll to see what they remembered.

24 Screenshot 2015-11-11 09.46.30

25 Screenshot 2015-11-11 09.46.13

Due to the success on the first question, I changed it up a bit with the second question.

26 Screenshot 2015-11-11 09.47.06

But I wonder now whether I should have started with the second question. If they could do the second question, doesn’t that tell me they can also do the first?

I’ve rearranged the polls to try that the next time I teach dilations.


We ended the lesson with a triangle that had been dilated. Where is the center of dilation?

And so the journey continues, with hope that noticing & noting will make a difference in what students learn and remember …


Posted by on November 17, 2015 in Dilations, Geometry, Uncategorized


Tags: , , , , , , ,

Cavalieri’s Principle

Geometric Measure and Dimension G-GMD

Explain volume formulas and use them to solve problems

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

How do you provide students an opportunity to make sense of volume formulas? I’ve written before about how we use informal limit arguments to make sense of volume formulas for the cylinder and prism and then Power Solids to make sense of volume formulas for the cone and pyramid.

Using a slinky, we briefly discuss Cavalieri’s principle.

Solids: equal height, cross sections for each plane parallel to and including the bases are have equal area.

What are the implications of Cavalieri’s principle here? (the two solids have the same volume)

1 Screen Shot 2015-11-12 at 10.43.38 AM

And here? (none, as the conditions aren’t met)

2 Screen Shot 2015-11-12 at 10.43.54 AM

When we get to the volume of a sphere, I’ve always told my students they’ll have to wait until calculus to make sense of the formula.


(I sneak in this exercise in calculus and wait for someone to notice the result.)

If I ever made sense of the volume of a pyramid or sphere using Cavalieri’s principle while I was in school, I don’t remember. (Surely I’m not the only one.) This year, though, I’m determined to do better. I’ve been saving Pat Mara’s TI-Nspire documents to think this through.

4 11-12-2015 Image001 5 11-12-2015 Image002 6 11-12-2015 Image003

How can you use these images along with Cavalieri’s principle to make sense of the formula for the volume of a square pyramid compared to the volume of a square prism with base and height equal to the pyramid?

When I got out the play dough to make more sense of the dissection of the cube, my coworker joined me. Our solid isn’t beautiful, but we get why the three square pyramids have the same volume and why one square pyramid will have a volume that is one-third of the square prism with base and height equal to the pyramid.

7 2015-11-11 16.03.19 8 2015-11-11 16.04.22 9 2015-11-11 16.05.49

What can you say about this square pyramid and cone?

10 Screen Shot 2015-11-12 at 11.51.51 AM

I like the visual image of seeing cross sections that aren’t congruent but have equal area.

Now for the sphere.

11 11-12-2015 Image004

What do you see?

On the left, a hemisphere.

On the right, a cone cut out of a cylinder.

What’s the same about the solids?

The sphere and the cylinder have equal radii and equal heights. Since the “height” of the sphere is its radius, the cylinder has height equal to radius.

What are the horizontal cross sections?

On the left, a circle. The radius decreases as the cross section slices go from bottom to top.

On the right, a “washer” (or officially, an annulus), where the outer radius is always the radius of the cylinder (constant) and the inner radius is equal to the height of the smaller cone formed with the inner circle of the slice and the center of the base (shown by similar triangles).

Dynamic geometry software shows us that the cross sections have the same area. Convince yourself that they do.

I convinced myself here:

Screen Shot 2015-11-12 at 12.53.37 PM

And then I looked at the next page, which allowed me to move the cross sections and see the similar triangles change.

13 11-12-2015 Image005

So what does this tell us about the volume of a hemisphere?

According to Cavalieri’s Principle, it has the same volume as the solid on the right.

How can you calculate the volume of the solid on the right?

Subtract the volume of the cone from the volume of the cylinder.

14 Screen Shot 2015-11-12 at 12.31.25 PM

And then what about the height of a sphere for which this hemisphere is half?

15 Screen Shot 2015-11-12 at 12.31.58 PM

The Illustrative Mathematics task Use Cavalieri’s Principle to Compare Aquarium Volumes could be helpful for exploring Cavalieri’s Principle. I’ve had it tagged for several years now. Maybe this will be the year we take time to try it.

And so the journey as both learner (student) and Learner (teacher) continues, with gratitude for those who share their work and those who are willing to pause their work long enough to learn alongside me …


Tags: , , , , , , , ,

Experiencing Dandy Candies as learner and Learner

Back in April, I had the pleasure of attending a CPAM Leadership Seminar with Dan Meyer on mathematical modeling, where he lead us through Dandy Candies. Dan wrote about this 3-Act recently here. I’ve used several 3-Acts with my students, but this was my first time to participate in one from a “lower-case l” learner’s perspective. I’ve read about “purposeful practice” and “patient problem solving” for several years now, and I know that I have some understanding of what they mean, but seeing them in action from the learner’s perspective is powerful.

A few things struck me during the seminar. We don’t do 3-Acts just for fun. (I knew this, but Dan made it very clear that this isn’t just about engaging students in doing something; it’s about engaging students in doing math. I’m not sure I’ve made that as clear to other teachers with whom I’ve discussed 3-Acts.) As Smith & Stein point out in 5 Practices for Orchestrating Productive Mathematics Discussions, there is pre-work for the teacher: identifying the math content learning goal for the lesson and then selecting a task that is going to provide students the opportunity to engage in that math content. Even with a 3-Act, where we let our students’ curiosity develop the question, we do have an underlying question that will engage students in the math content we want them to know. What they ask might not be worded exactly the same, and it might extend the mathematical thinking in which we want students to engage, but the math is there.

I have said before that I use technology to give every student a voice – from the loudest to the quietest, from the fastest to the slowest. When Dan solicited questions we could explore from the group, I was never going to volunteer mine for the list. (I am not criticizing Dan’s move here … just noting that I find it challenging, both as Learner and learner, to establish trust in a short session with participants that I’m likely not going to see again.) I *might* have participated had I been asked to submit my question somewhere anonymously.

And finally, I really like the opportunity that we had before each question to answer before performing any calculations. I’ve been working on providing this opportunity for my students, but it still isn’t automatic. I have to remind myself to ask students to use their intuition first. As I heard from Magdalene Lampert, “Contemplate then Calculate”.

We were working on Modeling with Geometry (G-MG) when I returned to class after the seminar last year, and so I tried Dandy Candies with my students.

1 Screen Shot 2015-11-10 at 11.22.56 AM

are the heights the same     1

what are the surface areas of the boxes    1

the similarity between how thevolume stays the same and te cross sections change1

Do all the solids have the same volume?    1

are the surface area and volume the same throughout the same changes?     1

do all the boxes have the same volume     1

how many cubes       1

what shapes could be made            1

how does the surface area change 1

same surface area?   1

could the volume make an equal ratio       1

whats the volume of each cube that makes each shape  1

how many different shapes can be made with those boxes        1

Do they all have the shme volume  1

Is the area of any gift formed by the candies the same? 1

5 Screen Shot 2015-11-10 at 11.19.07 AM

What do you *think*? Which package(s) use the least cardboard?

(No one answered more than one.)

3 Screen Shot 2015-05-05 at 10.05.13 AM

What do you *think*? Which package(s) use the least ribbon?

(No one answered more than one.)

4 Screen Shot 2015-05-05 at 10.05.19 AM

What do you *think* are the dimensions for each box?

6 Screen Shot 2015-05-05 at 10.04.43 AM7 Screen Shot 2015-05-05 at 10.04.52 AM

I enjoyed watching students use appropriate tools strategically while they were working.

8 2015-05-05 09.46.17 9 2015-05-05 09.46.19 10 2015-05-05 09.46.23 11 2015-05-05 09.49.39

And then I sent the polls again.

Which package(s) use the least cardboard?

(Two answered B and D.)

12 Screen Shot 2015-05-05 at 10.03.45 AM 13 Screen Shot 2015-11-10 at 11.20.42 AM

Which package(s) use the least ribbon?

(15 answered B and D; 2 answered B and C.)

14 Screen Shot 2015-05-05 at 10.03.51 AM 15 Screen Shot 2015-11-10 at 11.20.56 AM

16 Screen Shot 2015-05-05 at 10.03.26 AM 17 Screen Shot 2015-05-05 at 10.03.34 AM

Some mistook “better” for “best”, and others are apparently going to cut the candies in halves.

Modeling with Geometry G-MG

Apply geometric concepts in modeling situations

  1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
  2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
  3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Content-wise, students had the opportunity to learn more about modeling with geometry. And they were able to engage more steps of the modeling cycle than just computation.

18 Screen Shot 2015-11-10 at 1.03.48 PM

I am not *yet* writing 3-Acts, but as the journey continues, I am grateful for those who do and share …


Tags: , ,

Hot Coffee + Show Your Work

I’ve written about this lesson before, but I wanted to write again because of several observations from last spring.

I know that I need to find a way to provide students the opportunity to engage in math modeling more often and earlier in my geometry course. I’m having a hard time finding a way to do that. (Ideas for providing students the opportunity to engage in math modeling while proving theorems about congruence and similarity are welcome!) For now, we focus on modeling during the last unit of the course.

I decided last year to show students the modeling cycle from CCSS at the beginning of each lesson so that students would recognize what I am asking them to do differently and why I’m not giving them all of the information they need up front.

Our learning goals: I can model with mathematics, and I can show my work (leveled learning progression from Jill Gough).

0_0 Screen Shot 2015-11-01 at 5.51.40 PM

0_1 Screen Shot 2015-11-01 at 5.52.22 PM

Once we decided what questions to answer after watching Act 1 of Dan’s World’s Largest Hot Coffee Three-Act, students estimated responses.

0_2 Screen Shot 2015-11-02 at 5.30.22 AM 0_3 Screen Shot 2015-11-02 at 5.31.12 AM 0_4 Screen Shot 2015-11-02 at 5.30.57 AM 0_5 Screen Shot 2015-11-02 at 5.31.34 AM

And then teams made a list of the information they needed. I gave them information only as they requested it. Most teams realized later rather than sooner that they would need some type of conversion for cubic feet into gallons.

1 Screen Shot 2015-11-01 at 5.54.15 PM 2 Screen Shot 2015-11-01 at 5.54.39 PM 3 Screen Shot 2015-11-01 at 5.55.08 PM 4 2015-05-01 09.31.46 5 Screen Shot 2015-11-01 at 5.55.42 PM

When they decided they needed to know how much coffee a regular cup would hold, two of the girls remembered that the teacher with whom I share the classroom always had a cup of tea. They asked to borrow her cup so that they could come up with an agreed upon amount for a regular cup of coffee.

6 2015-05-01 09.45.32

At one point, a student asked whether getting the right answer mattered. I asked why. She and her teammate didn’t have the exact same calculation.

7 2015-05-01 09.52.04

It struck me that what we were really working on today was identifying a problem, determining what was essential to know, and creating a model to answer the problem. It’s not that the calculations aren’t important, but for this lesson, the questions were more important. By the time I got back around to that team, they had resolved their computational issue because of a conversion error. Even so, I’m glad I was asked whether it mattered that everyone got the same answer, as it helped shape how I launched our remaining modeling lessons.

8 Screen Shot 2015-11-01 at 5.46.22 PM 9 Screen Shot 2015-11-01 at 5.46.38 PM

And so the journey to provide students the opportunity to engage in all steps of the Modeling Cycle continues …


Tags: , ,

Structure, Flexibility, and Planning

I set up a recent lesson by asking students to deliberately practice SMP7, look for and make use of structure.

2 SMP7 Number #LL2LU Gough-Wilson

This practice requires us to make visible what isn’t showing. In geometry, that often means drawing auxiliary lines.

We don’t always see structure in the same way or at the same rate, so once you’ve found one way to solve the problem, I want you to also deliberately work on your mathematical flexibility. Find a second way to work the problem.

1 Flexibility #LL2LU Gough

I had 5 questions prepared, the last of which I learned about in Justin’s and Kate’s posts last year about a Five Triangles task. I’ve been thinking a lot this year about not only planning learning episodes but also planning ahead what instructional adjustments I’ll make based on the feedback I get from my students. In my planning, I struggled with which question to use first. Which question would you use first with your students?

3 10-25-2015 Image0014 10-25-2015 Image002 5 10-25-2015 Image004 6 10-25-2015 Image005 7 10-25-2015 Image006

Last year, I had the following results in the following order.

8 Screen Shot 2015-10-25 at 4.55.00 PM

After this first Quick Poll, I didn’t display the correct answer, asked students to team with someone else in the room, and sent the Poll again.

9 Screen Shot 2015-10-25 at 4.55.15 PM

After this Quick Poll, we had a student who answered 53˚ share his reasoning with the rest of the class so that we could figure out where the reasoning went wrong.

10 Screen Shot 2015-10-25 at 4.55.30 PM 11 Screen Shot 2015-10-25 at 4.55.43 PM

The class went fine. But I wondered what would have happened if I had started with a question that required the use of auxiliary lines (even though students struggled with the question that already had them drawn). So I tried that this year.

I could “hear” thinking and I could “see” productive struggle as students started out working the problem individually. Once they started sharing some of the ways that they made visible what wasn’t pictured, I saw evidence of SMP7. Because I had deliberately asked them to work on their math flexibility, they weren’t satisfied with only one way to solve the problem.

12 Screen Shot 2015-10-25 at 5.01.05 PM 13 Screen Shot 2015-10-25 at 5.01.18 PM

Many wanted to share their way with the whole class.

14 Screen Shot 2015-10-25 at 5.03.43 PM

They tried another one, and again, you could “hear” thinking. I didn’t even have to suggest individual think time to the class, as they naturally all wanted to try it by themselves first.

15 Screen Shot 2015-10-25 at 5.01.34 PM

I posed the folded rectangle problem, but the bell rang before students could really dig in to solving it. Maybe next year I’ll be brave enough to start with it, as the journey continues …

p.s. I’m currently reading Ilana Horn’s Strength in Numbers: Collaborative Learning in Secondary Mathematics, and before I was able to publish this post, I happened to read a section entitled “Turning Some Pet Ideas about Mathematics Teaching on Their Heads: Start with Challenging Stuff, Not Easy Stuff”. Her premise is that starting with easy stuff is inequitable, as students who get the mathematics quickly can take over the problem, and those who don’t miss out on the opportunity work with their team. Starting with challenging stuff levels the playing field for all students to contribute and learn.

Leave a comment

Posted by on October 26, 2015 in Angles & Triangles, Geometry


Tags: , , , , , ,

Conditional Statements, Contrapositives, and Indirect Proof

Towards the end of class a few weeks ago, we proved (using truth tables) that the original statement and its contrapositive are logically equivalent.

1 Screen Shot 2015-10-21 at 7.35.25 AM

I sent a Quick Poll to assess student understanding and was fairly pleased with the results. (Students had only learned which was which for conditional/converse/inverse/contrapositive/biconditional statements the day before.)

2 Screen Shot 2015-10-21 at 7.34.54 AM

We began to think about the implications of the original statement and its contrapositive having the same truth value in terms of proof, in particular, indirect proof. And then the bell rang.

3 Screen Shot 2015-10-21 at 7.35.37 AM

When we started class the next day, I sent a Poll specifically pertaining to indirect proof, wondering whether a) what we had learned last class stuck and b) whether they were able to transfer the statement-contrapositive-same-truth-value into how to begin an indirect proof.

Here’s what I got.

4 Screenshot 2015-10-01 10.01.25

What would you have done next?

I didn’t show the results of the poll to the students. Instead I gave them a different question from my stash, with more information.

5 Screenshot 2015-10-01 10.01.19

We talked about the responses, and then they tried an open response question.

6 Screenshot 2015-10-01 10.01.14

We talked about those responses, and then I sent back the first question.

7 Screenshot 2015-10-01 10.01.07

I’ve recently read Embedding Formative Assessment by Dylan Wiliam. Wiliam, along with countless others, suggests planning ahead a sequence of questions for a learning episode along with the instructional moves you’ll make based on the feedback you get from the students. What will you do next if very few of the students get it correct? What will you do next if half of the students get it correct? What will you do next if all of the students get it correct?

If most of my students had gotten the first question about indirect proof correct, I wouldn’t have sent the question with more information about indirect proof. I would have gone straight to the open response question.

I have the luxury of teaching the same classes and planning with the same teachers from year to year. Our stash of questions to ask during the lesson has grown based on responses from students from year to year.

And so the journey continues … teaming together to decide what questions to ask and what instructional adjustments to make, based on the feedback we get from our students.

1 Comment

Posted by on October 21, 2015 in Geometry


Tags: , , ,

Why I Use TI Technology

I was asked a few weeks ago why I use TI Technology in my classroom. I have never felt articulate when it comes to extemporaneous speaking, but I agreed to talk with the reporter because my experience has been that it’s good for me to have to justify what I’m doing in the classroom. Why do I continue to use TI Technology in my classroom when other free technologies are available for teachers and students to use?

A Google search of “technology speeds up life” results in about 142 million results in less than half of a second. What I find in my classroom, however, is that using technology actually slows down the pace.

In the midst of my usual rush to cover all of the required standards, when we use TI-Nspire Technology to explore a difficult concept, the questions that students ask slow us down. The platform that we use encourages students to ask questions beginning with “why does …” and “what happens when …” It allows the students to find answers to those questions interactively so that I am not the only expert in the classroom.

In the midst of my usual rush to cover all of the required standards, when I use TI-Nspire Navigator to send students a Quick Poll to check for understanding, their responses sometimes let me know that we need to spend a little longer “attending to precision”. Seeing that their response doesn’t get marked correct up on the board in front of the class is not the same thing as me writing or saying the correct answer and having students individually check their work. And so we look at their Quick Poll responses together, letting the students determine whether or not the answers are correct or incorrect, letting the students determine what error was made to produce an incorrect answer, “critiquing the reasoning of others”, learning from both correct and incorrect responses.

Most importantly, using TI-Nspire Navigator gives every student a voice to make their thinking about mathematics visible – not just the loudest student, and even the quietest student … from the one who answers quickly to the one who needs more time. I no longer have to rely on their polite nods to determine whether my students are “getting it”. Their responses to Quick Polls let me know whether they are getting it, and give me the data I need to determine what instructional adjustment to make next in the lesson. Using technology eases the hurry syndrome, forcing me to pay attention to the questions students have and allowing me to assess their progress continuously.

Our 9th graders are 1:1 MacBook Airs for the first time this year. We use TI-Nspire Navigator for Networked Computers with them, and we continue to use TI-Nspire Navigator for handhelds with our other students.

Quick Polls give me information on where students currently are in their thinking.

The answers can be multiple choice:

1 screen-shot-2013-05-11-at-4-56-13-pm

If your learning goal for a lesson is for students to determine the center and radius of a circle by completing the square, what would you do next after the results above?

We had our students convince each other of their answer and sent the poll again:

2 screen-shot-2013-05-11-at-4-58-21-pm

Multiple select:

3 Screen Shot 2015-10-20 at 5.35.15 AM 4 Screen Shot 2015-10-20 at 5.35.30 AM

5 11-05-2014-image004

An equation:

6 11-05-2014-image005 7 screen-shot-2014-11-05-at-2-36-18-pm 8 screen-shot-2014-11-05-at-2-36-56-pm

An ordered pair:

9 Screen Shot 2015-10-20 at 5.43.50 AM 10 Screen Shot 2015-10-20 at 5.43.58 AM

A point:

11 Screen Shot 2015-10-20 at 5.47.41 AM 12 Screen Shot 2015-10-20 at 5.47.58 AM 13 Screen Shot 2015-10-20 at 5.48.08 AM

Or an expression, providing students the opportunity to attend to precision:

14 Screen Shot 2015-10-20 at 5.57.32 AM

All of the data that I collect from my students during class is stored in a Portfolio, which is helpful for seeing who needs extra support and/or enrichment.

14_1 Screen Shot 2015-10-20 at 8.59.00 AM 14_2 Screen Shot 2015-10-21 at 5.36.22 AM

Class Capture gives me information on where students currently are in their work, allowing me to monitor, select, and sequence for whole class discussion.

15 6-screenshot-2015-09-04-09-25-2915_1 15_2

Making a student the Live Presenter displays the student’s device in real-time to the whole class. (With TI-Nspire Navigator for Networked Computers, what the student displays isn’t limited to TI-Nspire … a student can display and interact with any application on the computer.)

16 screen-shot-2014-09-10-at-9-34-12-am 17 screen-shot-2014-09-10-at-9-38-41-am

Interactive TI-Nspire documents allow us to explore mathematics using graphs, geometry, data and statistics, data collection probes, calculations, and interactive notes.

18 03-02-2014-image002 19 03-02-2014-image001 20 03-02-2014-image003

21 10-20-2015 Image001 22 10-20-2015 Image002

23 06-17-2015 Image005 24 06-17-2015 Image006 25 06-17-2015 Image007

With TI-Nspire Navigator for Networked Computers, I can send (and collect) all types of documents, not only TNS documents.

26 Screen Shot 2015-10-21 at 5.20.27 AM

All in one piece of software.

In addition, the T3 community has taught me and continues to teach me more than I ever thought I could learn about mathematics and pedagogy and the appropriate use of technology for improving student understanding of mathematics. Gail Burrill and Tom Dick always make us think about whether we are “using technology as a tool for calculating or as a tool for deepening student understanding of mathematical concepts”. I am (admittedly) a proud member of that community.

I have no agenda to force a certain technology on anyone. Even so, it has taken me a while to find my voice as part of the #MTBoS. Many speak against TI so loudly that I sometimes wonder whether others can see that what I share through my blog and tweets is a journey of learning, teaching, and questioning that is platform-agnostic. I use what I have access to use and what makes my students’ thinking visible. I hope that others are doing the same.

And so the journey continues … full of hope that our collaboration and conversation can transcend platform and focus on deepening student understanding of mathematics.

1 Comment

Posted by on October 21, 2015 in Professional Development & Pedagogy


Tags: , , , ,


Get every new post delivered to your Inbox.

Join 2,162 other followers