See here for one such example. Having learned about tangent lines to graphs, what if you are given only a tangent line, and no graph? In the sketch, we have a function f(x), and a tangent line through a point (a,b) with slope f(b). Move point A around – what do you you wonder? What students ask when the see the tangent line free to move around is enlightening and inspiring for me as a teacher. Students really do re-invent calculus in their own way through their own curiosity.

Understanding connections between graphical, numerical, symbolic, and verbal representations is essential in all areas of mathematics. In the study of calculus, symbolic language, in particular, is a key to opening doors of understanding. Without connections, however, it may be difficult to know what lock the key fits.

Understanding also depends upon a foundation of experience. If learners’ experiences are limited, they may not be able to “see” past the next question. When learners can “see” more of the terrain of ideas from which a particular idea arises, they are in a better position to construct meanings that will allow them to travel further across the terrain.

For example, students can shed light on the fundamental ideas of calculus from by investigating limits, derivatives, antiderivatives, and integrals during the first week of calculus. When students develop a vision for calculus, they seek out the connections that enable them to develop deep understandings.

Conducting investigations as a means of developing precise definitions puts students into the same shoes as Newton and Leibniz. While these two mathematicians are credited with inventing the calculus, they did not use the definition of limit that we use today. The precise definitions contained in textbooks today arose, over years of study, from the intriguing questions that Newton and Leibniz asked but did not fully answer.

The students of today face the same questions and cognitive obstacles that faced those who developed the calculus, but we ask students to face them over the course of two semesters. The temptation for the teacher may be to “make things clear” for the students. Sometimes a shortcut, however, is not worth the sights the students will miss along the way.

With Sketchpad, students can experience calculus through direct action on dynamically defined mathematical objects. Using custom-built construction tools, students’ ideas become actions, and their actions become ideas. By experiencing calculus in this way, students can begin to make things clear through their own activity.

In this institute, you will take your own journey of exploration of the terrain of calculus. On the way, you will likely meet some familiar friends, but see them in a new light. You will walk in the steps of your students, to whom the terrain is unfamiliar, while using the shoes of Newton and Leibniz.

In this institute, you will see how using Sketchpad gives students a map for their journey through calculus, and a sense of where they are going and why. The point of their journey should not simply be “to arrive” – but to learn from the journey itself. We hope your journey this week is that kind of journey, and that you will be motivated to return to see the sights in new ways again and again.

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I see a lot of these structures as metaphors for experiences learning math in and out of the classroom. In any CYOA, regardless of the specific structure, experiencing failure and then going back to make new choices is an integral part of the experience. The role of productive struggle, and that making mistakes in math class is a good thing, has certainly received renewed attention in the last few years. I’ll expand upon the metaphor as I post the elements of the talk.

]]>Davis, G. E. (1996) *What is the difference between remembering someone post a letter and remembering the square root of 2?* It was in the Proceedings of the 20th Conference for the International Group for the Psychology of Math Education.

I’ve been thinking about it ever since.

You can find the whole article here.

I mention it here because that article made me, and still makes me, think about the importance of students remembering math, to quote Davis, as more than “a rule-driven, bloodless, passionless, activity, situated nowhere in time or space.”

He writes, “Why, however, should a young student’s mind actively engage with mathematics when they don’t yet know what it is about? It is not, after all, and every day subject that most people talk about as they do the weather, their health, or the activities of their friends and neighbors. But perhaps it could be…”

In this article, Davis writes about the importance of of developing mathematical imagery that make abstract concepts something more “real”, and creating memories that situate ideas in time and place, with a sense of meaning. The idea that “abstract knowledge, once obtained retains within it elements of the original context” seems likely a reasonable idea but one that I think we do not pay enough attention to.

Davis references a 1951 book by E.J. Furlong, *A Study in Memory*, stating that to “recollect, reminisce, and retrospect” – with imagery – is essential to mathematical thinking.

It’s this development of a “rich set of images” about mathematics that interests me. These images are more than pictures we might draw on paper, but the actual contexts in which we have experienced mathematical constructs. The students who have participated in the Random Walk lesson have a certain “lived experience” and imagery associated with coins flips and random behavior because of the experiment. I have found that they refer to their actions in the lesson years later when they talk about randomness.

I don’t claim that makes this lesson the “best” lesson for teaching this topic. But I do argue for the importance of viewing any math concepts as part of a story – a story in which each student is a *character*, not just a *reader*. The Random Walk lesson has been particularly effective in engaging students as protagonists of a personal story of math, one that goes beyond “I remember I learned *about* that math topic in 7th grade, but that’s about all I remember.”

There are certainly lots of ways to engage students, but I think those that promote students’ development of the “episodic memories” of which Davis writes have a particularly important effect on students’ learning of abstractions. Math *can* be something we talk about in the same way we talk about much of the rest of our lives.

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A student takes a random walk on a number line determined by flips of a fair coin: one step forward with a flip of a heads, and one step back with a flip of tails. After students propose questions about an individual’s random walk, the whole class, joined by other students if possible, conducts a simultaneous random walk. Students arrive in various locations on the number line and then walk into columns to create a human bar graph that records their final location. Students then investigate the theoretical probabilities involved in the random walk.

After the initial random walk of one student, I have asked students to record what they noticed and what they wondered. Here are a few of their thoughts:

- How often will I get heads or tails?
- Where will I arrive after a certain number of flips?
- How long will it take me to get to the other end of the room?
- What are my chances of getting to the other end of the room?
- It would take forever just to walk a couple of feet!

I tell students that in order to investigate these and other questions we are all going to do a random walk at the same time. Before we do this, I again ask,”what would you wonder?” Students are interested in where everyone would end up – and the walk itself would look like. One prediction is that everyone will end up back in the middle because we all have an equal chance of getting heads and tails.

After having students record their thoughts about where they could end up, and how many people might end up at each location, we go out to a hallway or other suitable location. Everyone lines up, and we do individual walks all at once (on cue, one flip at a time.) Students have a calculator to generate their motion forward or back, or we have already flipped coins and written down our individual directions. Here’s an example of what it looks like “from above,” after each step, for 5 flips…..and after the 6th flip; then we march together to form a human histogram. 6 flips is a good number to make an interesting histogram.The result is a surprise for many students – and something they consider worth explaining. Usually without my even asking, students return to the class with lots of noticing and wonderings. Why did the largest (but often not the majority) of people end up back in the center, at 0? Since there are 7 possible locations one can arrive at, why is the probability of reaching each final location not 1/7? I encourage students to investigate and explain the results using whatever approach they wish.

Many students turn to making a systematic list of possible sequences of coin flips. (Below is a list for 4 flips.)

I encourage students to create a list of possible flip sequence for 2, 3 and 4 flips. In creating such lists, there is often discussion about whether a sequence like TTHH is “different” than that of HTTH, since they are both “2 heads and 2 tails”. I have found the context of walking (“I had a different walk but ended up in the same place”) helps students think about why each of these possibilities should be listed separately when thinking about the probability of arriving at any of the possible locations.

After the students create lists organized like the one shown above, they notice that the list looks similar to the human histogram they had created – but there is more that students want to know. They have a lot of fun doing the walk and watching it happen, but most importantly it leaves students asking more questions.

To be continued…

]]>Being a Geometer’s Sketchpad enthusiast, I had been playing around with a sketch that allowed a point to take a random walk on the number line. The students quite liked watching the point dance back-and-forth, sometimes off the screen to the left or right, and had lots of questions about random behavior.

So I thought, for reasons I can’t currently recall, that it would be cool to have the whole class do what that point was doing, en masse, out in the hallway. Students would end up at different locations on the number line, and then we could form a human histogram of our final locations by stacking ourselves together down at the end of the hallway.

This was a big hit. The students enjoyed watching the dance (a pre-made sequence of coin flips or a scientific calculator generated our steps to the left of right) and had even more questions. The lesson, for 7th graders, centered around determining probabilities and generating the different possible ways of landing on any particular location in the number line. It was definitely memorable the next year when we followed up with more analysis of the probabilities and got into Pascal’s Triangle.

When I presented the lesson at NCTM in 2010, an attendee went home and tried it with a group of 100 teachers at a summer workshop.

Last year I decided to bring this lesson to my new school, and made some modifications – the students stand shoulder-to-shoulder and walk forward and backwards rather than left or right. This allows them to more easily see the motion of their classmates and feel like they are moving like a point on a number line. I submitted this lesson for the National Museum of Mathematics’ Rosenthal Prize – and it won. (!) I have found the lesson to be very engaging for students and hope to see more teachers try it out.

Here’s how the lesson starts:

*We’re going to go for a walk today. But it’s going to be a funny kind of walk… we’re going to let coin flips be your guide. *

Select a volunteer walker and another student to flip a coin. With the walker standing in a suitable location, explain to the class that the student will take one step forward when the coin lands heads, and one step back when the coin lands tails. With each flip, have the volunteer take a step forward or back. Repeat 6-10 times to interest students in the randomness of the student’s motion.

*If you were feeling random one day, and were to take a walk with coin flips as your guide like our volunteer just did, what would you wonder about? What would you be curious about?*

More about the lesson in my next post!

]]>When we talk about understanding, we often use words associated with vision and sight. Yet there is so much to the concept of “understanding” that does not fit the conventional definition of something that is literally “visible.”

In my first year of teaching, 23 years ago, I asked my students to make “magazines” called “Math Illustrated.” The purpose: tell your story. What do the ideas we are learning “look like” to you? Use words, pictures, graphs, equations, artwork, poems – whatever else strikes you as useful. You’ve got your own story – just because we were all in the same room today doesn’t mean we all experienced the same thing. The world wants to read about you story! (Or at least I do!)

This request of my students grew out of my own experiences with learning mathematics up to that point. I found that when I tried to make sense of math, I engaged in dialogues – trading words, metaphors, images with others. I tried to “see” things from others’ perspectives. I would “negotiate” meanings, (Is this what that means? OK, I’ll settle on that for now.) More often than not, I would come back and reconsider my decisions based upon new information.

Sharing thoughts with others is, for me, a central part of mathematics. Sometimes, the sharing is only with one’s own self. I trade thoughts with authors past and present, though their written records. I try to see an author’s motivations, trains of thought, and goals, in light of my own experiences. When I share ideas with colleagues or friends, we talk, write, draw, question each other. We look at each other and ask, “Do you see what I mean?”

Conversing with students, I try to initiate and sustain dialogues focused on building shared understandings, viewpoints, and perspectives, I seek to find out what the mathematics “looks like” to that student, in light of their experiences. I wonder if I can I see it the way they are seeing it. I try to make math an ongoing conversation where we’re all looking at something (and maybe not always the same thing!) and have lots of share about what we are “seeing”.

I’m never been quite sure that anyone else “sees” an idea in the same way as another person. That, to me, makes things really interesting! That will be the goal of this blog – to illustrate stories and experiences of learning math.

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