I recently prepared a series of number explorations for a 2nd grade classroom. The students in this classroom have been working with base ten concepts and place value and are now beginning to compare numbers. In this post I'll share a menu of activities that can be used to support deeper understanding of these concepts. The goal here was to help students reason about the relative size of numbers and quantities without relying on an algorithm to compare numbers. I've also tried to help students build some bridges between various representations of numbers.

1. We started with Andrew Stadel's Estimation 180. While many of his explorations might be targeted at an older audience, I found the following extremely well-received in this 2nd grade classroom.

First huge moment: seeing licorice on the big screen at 12:45 in the afternoon.
Second huge moment: getting to guess and seeing those guesses on the board.
Two equally huge final comments: "I was sooo close!" vs "Wow, I was not even close!" These are both satisfying to me.

Students had opportunities to consider the relative size of numbers as each classmate shared a guess. We discussed our smallest and largest guesses and any that appeared more than once. It was quite evident that students were wondering whether they should guess a really big number, and if so, what that number would be in this scenario.


2. We followed this up with some number line estimation. I used the cards below with the students seated on the carpet for a brief number talk. Along with estimating the location of the red dot, students answered the question, "How did you decide?" This line of questioning encouraged students to use the numbers given as landmarks and do some broad thinking about tens and hundreds.

3. Next we pulled from the deep constructivist work of John Van de Walle. I was first introduced to John Van de Walle's work in my teacher credential program back in 1996. These activities are great work for both teacher and student and make mathematical reasoning about number the focus.

4. I pulled this next task from Illustrative Mathematics as a progressive step toward more challenging estimation with number lines. I drew a long number line across the board at the front of the room and labeled every hundred up to one thousand. Students teamed up and together discussed and decided where to place their number.

5. The final selection in this menu of activities is from the National Library of Virtual Manipulatives. I used the Practice mode with one dot in which you are asked to drag the dot onto the number line to match the value shown. If the location you choose is not close enough to the actual value, you are asked to zoom in and be more precise.

Students love these problems. They're neat, there's one possible solution, and Hey! - they look like trees.

No need to discuss prime factorization, because a) students don't understand those terms, b) what students really want to know is how to solve the problem, and c) it's much easier (and cuter) to say, "Just make a factor tree.”

What we have effectively done here is reduced one of the most fundamental theorems in all of mathematics to a Christmas tree ornament. This isn't the first time we've done this. We define prime numbers for students, we tell them the value of pi, we calculate the sum of the triangle's angles for them. We sell all the secrets, reveal all the magic, and spoil all the endings. 

Granted, we've all tried to play it up big in the classroom in an attempt to generate some enthusiasm. We soon discover that not all students are as interested in significant mathematics as we are. I'm not saying we should attempt to surprise students for the sake of enthusiasm levels. Besides, students who are struggling to understand something may not feel any sense of surprise. Significant mathematical surprise requires solid intuition - enough to appreciate the differences between intuition, expectation, and fact. But if I can get a "Wow, that's kinda cool!" or a "I didn't expect that!" from a student, I'll take it.

So here's my attempt to return to students some opportunity for discovery:

1. Make a list of the first few primes: 2, 3, 5, 7, 11. Give students a composite number, say 6. 
   
2. Now ask, "Can you make the number 6 using multiplication and only numbers in this list? You can use a number more than once."
3. When they come back with a solution, point out the obvious. "You didn't use any other numbers? Only primes?"
4. Give a few more to try: 12, 14, 20, 45, 28. "See if these numbers are possible." Or ask students to volunteer composite numbers between 1 and 50.
5. Ask, "What other composite numbers do you think you can make?"

What are students wondering now? Are they thinking about which numbers are possible? At this point do they expect that every composite number can be written as a product of primes in a unique way? Do they understand that they have just stumbled upon the Fundamental Theorem of Arithmetic that was first proven by mathematician Carl Gauss in 1801? I can't expect surprise, or even appreciation of this discovery. But students are making conjectures and testing them as thoroughly as they can, and that's worthwhile.

George Pólya gave ten commandments to teachers. This seems fitting:

I'll agree that a factor tree is an efficient tool for grinding out prime factorization. However, it is unfortunate to see absolutely no connection between the tool and the discovery. Why not wait until students see the need for such a tool? When discussing ways to reach the number 48, I have seen students start with factors 6 and 8, then represent these using primes. These students needed a way to organize and keep track of this process and were able to invent their own tree-like structure. Save the trees for art class and let your students do some real thinking. You'll be helping to spread the joy of discovery. 

Here's a short video introducing a new course I'm offering with the Center for Professional Development.

In the first lesson, students were invited to create a new game based on the four-square pieces used in TETRIS. Using five identical squares, groups explored possible arrangements and clarified guidelines for piece creation along the way. Part 2 continued as students kept track of possible pieces by cutting them out of a 1-inch grid on cardstock.    

As collections grew, quite a few groups had several pieces that were duplicates. I anticipated this, and so prepared a few slides to challenge their spatial reasoning and sharpen their skill at finding these duplicates.

After groups were convinced we had found all 12 unique pentominoes, I reminded students that what made someone really exceptional at TETRIS was an understanding of how the game pieces fit together. I explained how clearing lines was related to making rectangles and challenged them to practice making rectangles by putting together various pentominoes. I used this handout to help structure their exploration.

I suggested that students treat this part of the lesson as a tutorial, similar to the TETRIS tutorials. It’s not the real game, but exploring how the pieces fit together would certainly support their gameplay. We spent about 10 minutes on this task before getting to the real deal.

I asked students, “What do we do with these new game pieces? How does the 5-square game work?” These questions led us to take a closer look at what makes the TETRIS game tick. Together we brainstormed some of the game features:

1. What’s the goal? Fit pieces together, clear lines, and keep the screen from filling up.
2. How does game play work? Player vs. Computer
3. How are pieces chosen? Computer chooses randomly and presents next piece to player.
4. What’s the game board like? 10 x 18 grid.
   

Students immediately pointed out the constraints we were dealing with that would prevent a TETRIS-like experience. We didn’t have unlimited pentominoes and clearing lines wasn’t possible with physical pieces. I decided to use this opportunity to present them with the challenge of actually putting together a new game. Here’s what I said:

In order to create a working game, you’ll need to come up with precise responses to the four questions we just discussed.

Groups got to it and after a few minutes a whole class discussion yielded these results:

Answers to questions 1-3 were worked out pretty easily, but defining the game board size left us stumped. We ended up creating several sizes and ran some game trials.

Gameplay using these two boards led us to think about how many squares we really needed.

From their work building rectangles during the tutorials earlier in the lesson, students knew it was challenging to create a rectangle using all of the 12 pentominoes. And from their gameplay using the two trial boards, they knew there needed to be some extra squares for those times pieces didn’t fit together perfectly. Thus, a board with exactly 60 squares was ruled out. We settled on trying an 8×9 and an 8×8.

Based on these results, most groups decided to use the 8×8 board for future play. A few explained that the extra row on the 8×9 board made play easier and they preferred it over the 8×8. Overall I found this activity to be a motivating context for student use of mathematical practices and an inexpensive way to get a great game in kids’ hands during the last few days of school.

I started this lesson by telling kids they would be working together to create a new game. I asked them if they’d ever played TETRIS. The response was mixed. I don’t know when this game hit its peak, but I remember wasting tons of time on it and practically having a heart attack as pieces didn’t go where I wanted them to and the screen filled up. I wanted to use that sort of hysteria as classroom fuel, so I went over to tetris.com and pulled up some of the interactive tutorials. These are quick, direction-led tutorials that allow you to get a feel for the game. I randomly picked a few students to give these a shot while the class watched.

The first tutorial involves moving a piece from the right wall to the left.













Piece of cake. Then we rotated a piece and cleared a line. But then we had a few pieces coming one after another and had to move rather quickly.

A student came to the front and got started. When she got to this point, the class was screaming at her.

ROTATE! ROTATE! ROTATE!

It was great to see such a need for rotation.






After we discussed the rules and some observations they made about the pieces, I presented them with the new challenge. What if we used five squares, rather than four, to make these game pieces? How would it change the game? How many unique pieces would we need? Students made some conjectures and most agreed that we’d have more pieces than we used in the TETRIS demos.

I gave each student five squares and asked them to build a piece they thought should be included in the game. After about a minute I asked students to get up, leave their pieces on their desks, and do a quick gallery walk around the room to view the other pieces. After they returned to their seats I asked students if they had seen any pieces they would include in the game. I also asked them if there were any they thought should be excluded.

Several students weren’t sure about this one:

But others pointed out that it made sense because of this one:

Then the attention was drawn to this one:

Most students thought this should be excluded. I told them that if they wanted to exclude it, they would have to come up with a new rule to clarify the guidelines for piece production. Here were their first attempts:

“A square can’t be in the middle of two other squares.”
“A square can’t be on a crack.”
“It has to allow others to fit without gaps.”
“It can’t be over half a square.”

With each of these attempts, I tried to show how these statements eliminated viable pieces they had already created. I pressed them regarding how very imprecise their rules were, and encouraged them to think about what each square had to do, rather than what it couldn’t do. From this discussion, we came up with:

Use five identical squares. Put the squares together so that each square shares an edge with at least one of the other squares. Turns or flips are duplicates and should be excluded.

 Satisfied with the guidelines, students set off to explore possible pieces.

In the next lesson students will be cutting these out and exploring ways they fit together. Some will most likely find pieces that slipped past the guidelines, like this one:

Recently I participated in City Summit 2013, a local community-based effort to connect people and their city in new ways, and in the process, foster a sort of re-imagining of what communities might look like. Keynote speaker, John Perkins, recounted his own personal story growing up as the son of a sharecropper in Mississippi in the 1930’s. Through a series of difficult and painful experiences Perkins became conscious of the racial and social injustices faced by African Americans in Mississippi. These experiences fueled his later involvement in the desegregation of Simpson County schools, as well as his work with the broader civil rights movement.

In his keynote, Perkins described the empowering nature of education:

“Education allows us to subdue our environment rather than be subdued by it.”

The Algebra Project

Robert Moses began working with civil rights activists in 1960, and in 1982 founded the Algebra Project, a foundation committed to establishing quality mathematics education for all children. The aim of this foundation and its current work is to increase access to and understanding of algebra in underserved communities, with the expressed conviction that this understanding has significant impact on students’ social and economic futures.

In discussing Moses’ convictions, UC Berkeley Professor Alan Schoenfeld states:

“Who gets to learn mathematics, and the nature of the mathematics that is learned, are matters of consequence.”

In a 2013 interview, Moses described the pivotal role algebra plays in a society increasingly dependent on technology:

Host: “Talk about algebra and what makes it important in general terms. Is it a skill that needs to be acquired for its own sake, or to give students a framework for thinking about other things in different ways?”

Moses: “The information age…has ushered in a quantitative literacy that has put algebra and logic as a necessary literacy for our democracy.”

KQED interview Bob Moses 2.6.2013

On Equity

During the 2013 NCTM Annual Meeting held in Denver, CO, Uri Treisman delivered the Iris M. Carl Equity address. His comments called attention to the empowering nature of mathematics education:

“Mathematics is the biggest determinant in upward social and economic mobility. We need to rebuild our education systems so they allow students to advance. Schools are places where we produce citizens with deep commitments to democratic ideals.”


Mathematics as a tool


We use mathematics as a tool to make sense of and understand the world around us. We need mathematics to help put events and trends into perspective, to look rationally and reasonably at aspects of these events that may not be apparent on the surface. Mathematics helps us go deeper.

Common Core Mathematical Practices emphasize critical thinking as students make sense of mathematics, and as teachers work hard to ensure that students learn to solve real world problems. But what kind of problems do we hope they solve? How will students learn to recognize problems that are in need of solving, and from what perspective will they approach these problems?

We want our students to ask important questions: What’s a reasonable wage? What figures might constitute discriminative behavior? Why are certain communities underserved in terms of assets or resources?

The question here is whether it is enough to deliver a rich mathematics instructional program, or whether we should also strive to help students use mathematics in meaningful ways.

Think about some of the enduring concepts students explore during their school study of mathematics:

number
, counting, 
patterns, 
structure, 
measurement
, data, 
change
, variable, 
function, 
statistics

There are, of course, more. Now think about a current issue or problem that needs attention in your local community, city, or state. You may even want to think globally. Here’s the question:

How many of the above concepts are required to think critically about this issue and understand it? Better yet, how many of these would play a role in any sort of solution to this problem?

Could it be that while students have been asking “When will I ever have to use this?” we’ve been giving many of the wrong answers? Rather than listing jobs that require math skills, perhaps we should revoice that question for students:

Oh, you mean “When will your mathematics understanding intersect an opportunity for meaningful change?”

The world needs mathematical thinkers.