How does a mathematician see the world?

And how does a math teacher help learners to see the world through a mathematician's eyes?

This was a major challenge during the past semester in my Intermediate Algebra sections. Many of the students came to class expecting me to show them a procedure that they would practice until test time - when they would reproduce the procedure and promptly forget it. Nearly all of the students had seen the Intermediate Algebra content in high school (linear, quadratic, and exponential functions) but it hadn't stuck.

This was the problem. They had seen the content. Now it was time for them to use it to see the world. So I shared pictures and videos and asked them to look at them through a mathematical lens.

At one point, a student said, "Just once, I wish I could see the world through your eyes." Exactly! Unfortunately, they were often so afraid of making a mistake, of breaking the mathematical glasses, that they were not willing to even put them on. They did not know how to be playful with the math we were exploring.

So I introduced them to Yes, And ...; this is a problem finding activity that I developed using a well-know improv game where participants accept and build on the ideas of their partners. Here are the instructions for my version:

• Provide a mathematical context (often pictures or videos) but without any identified problem;
• Pair up the students and find a fun way to identify Student A;
• Student A picks one of the contexts and finds a problem to solve;
• For one minute (this is usually enough time to get started without completely solving the problem), Student A talks through his or her thinking while Student B writes as much as possible down on a piece of paper;
• After one minute, the students switch roles. I say, "Yes, and ...," to signal the switch and to emphasize that Student B ought to build on the work that was already done.
• Student B thinks out loud for one minute while Student A records the thinking on the same paper.
• After one minute, I say, "Yes, and ...," and the roles reverse again.

Yes, And ... can go on as long as the teacher wants. I found six minutes (three rounds) to be about right the first time we played the game. One student submitted this to demonstrate her engagement with the task.

I cannot claim for certain that Yes, And ... changed my students' view of mathematics or helped them to see the world through a mathematical lens. What I know is that for six minutes they played with math. It's a start.

Where was I using the Process Standards?

MTH 110 is an Intermediate Algebra course that explores topics typically found in Algebra I and II. Most of the students have seen the content before but for some reason it did not stick (nobody's fault, just a fact). Instead of simply re-teaching the topics, we are using the content to extend our understanding of what it means to do mathematics.

One of the ways we are attending to the mathematical process is by making our thinking visible is through Metacognitive Memoirs. Because this is a new approach to many of my students, I spend a good portion of Flight School (the first three weeks of the semester) developing the idea of how we can go beyond showing our work. This past week, I facilitated a workshop that demonstrated one way to share our thinking.

Schema Activation: Predict how many small cubes are in Step 43

from Visual Patterns

Focus: NCTM Process Standards

As I share my thinking, please keep track of where I am attending to the bullet points associated with these processes

Activity: Metacognitive Memoir Demonstration

Reflection: Where was I ...

• Problem Solving
• Reasoning
• Communicating
• Connecting
• Representing

It might help to see my notes.

Please share your thinking (where I made the processes visible and opportunities I missed or messed up) in the comments.

What happens when I add this?

My preservice elementary teachers are just finishing up a unit on patterns. Coming into the unit, one thing some of them struggled doing was writing an explicit rule for a pattern reflecting linear growth. In my opinion, this represents a gap in experience not a lack of ability. Since all of them have had some background in Algebra, the question is what experience are they missing?

Coincidentally, a post by Dan Wekselgreene on Linear Patterns in Algebra 1 showed up in my Twitter feed this morning. (Thanks MTBoS!) It is a good example of immersing learners in different representations and supporting them in making connections that can result in deeper understanding of linear relationships. I had my students do something similar (see the example on the right) but for some it was not enough. There was still something missing. It is for this reason that I add one more representation (an idea I learned from literacy instruction) - a recount.

From Mooney's (2001) Text Forms and Features:

Recounts: Why? To give a sequential and detailed account of an incident, a series of incidents or a conversation

What? A written record of recall of events, with attention to sequence and accuracy, and often to detail

Doing a detailed account of each step provides learners with an added experience for seeing the structure of the linear pattern. A recount is demonstrated in the think-aloud provided below.

This seems to provide many of my learners with an experience that was lacking from previous math classes - a missing piece to the puzzle that bridges some cognitive gap. However, I am also aware that they have had experiences that the typical freshman in Algebra 1 might be lacking. What do you think, would adding a recount to your linear pattern unit help?

What goes here?

I look forward to learning with you.

The day before a teaching observation I send out an email confirming that I have the correct details and reminding the teacher that I will need an action plan beforehand to focus my attention. I try to end each of these emails with the statement provided at the beginning of this post. It serves as a reminder that the observation will be a learning opportunity for both of us and not just a dog-and-pony show.

Sometimes, I take for granted my role as a learner in these experiences. I was reminded of this during a recent observation. The teacher wanted me to focus on whether or not she was providing adequate support to students as they were learning how to multiply and factor polynomials. This was in an eighth-grade class and they were halfway through the unit - just wrapping up multiplication of polynomials.

The lesson went well and the students were engaged in what Fisher and Frey call Collaborative work. This entailed a few items that provided students practice multiplying polynomials and a worksheet called Polynomial Puzzler that the teacher had modified from a lesson found on Illuminations. The teacher went over the instructions,

Fill in the empty spaces to complete the puzzle. In any row, the two left spaces should multiply to equal the right-hand space. In any column, the two top spaces should multiply to equal the bottom space,

 and demonstrated using the first puzzle.

As I looked through the entire worksheet, however, I identified an area I thought would give the students trouble; there were places where the four entries in the upper left were missing entries. This meant the students would need to factor some polynomials in order to complete the puzzle. The teacher had not provided the necessary support for student success. I made a note to talk about this during the debriefing.

Sure enough, after students had completed the first puzzle and multiplied what they could in the second puzzle, many started to ask, "What goes here?"

The teacher responded with something like, "That's a great question. I guess I didn't give you enough support." She pointed to the upper right corner and said, "We want to find out what goes here. In other words, what times this {pointing at (-15x+3)} will equal this {pointing to the lower right corner}? Okay?"

I thought to myself, "No. Not okay. They need to know how to factor." But the students seemed satisfied and went on about their work. And the surprising thing was that they were okay. In fact, they were better than okay. They were amazing.

The students at the table nearest me began looking back at the work they had already done and sharing what they noticed. "Look. -4x+10 divided by 2 is this one {pointing at -2x+5}," one of the girls exclaimed with more enthusiasm than I usually see in math class. The group then began talking about dividing the polynomials to find the missing entries. Sometimes they tried using guess-and-check to identify what was missing. The main point for me was that they did not just give up.

They seemed to be embracing the struggle that so many students in math class are determined to avoid. And they weren't alone. I heard several ahas from students seated in different groups around the classroom. I do not know why this class acted so differently than others I have seen. The cooperating teacher is a former GVSU graduate, so I would like to think that the learning environment had something to do with it. Or maybe it was the fact that it was a puzzle and not homeWORK.

After the lesson, I asked the teacher if she had anticipated any problems. She had thought that the instruction might be confusing. When I pressed about the factoring, she acknowledge that it could have been a problem but that she thought it was actually good that they did not know exactly how to do the puzzle our way because then they would obsess about doing it right. "Besides," she said, "I just wanted it to foreshadow factoring. Now they're ready for the next lesson."

I love my job. I learn something everyday. Even on those days when I think I'm the teacher.

Did you get anything good?

For years, I have tried to write first semester exams using a holiday theme. Even as a middle school math teacher, I set problems in the North Pole or around different traditions celebrated this time of year. Given a recent #mathchat on holiday mathematics, I thought I would share some of the problems here. Consider it my gift to you. Happy Holidays!

On the twelfth page of the test my teacher gave to me, twelve days of giving.

Use the lyrics from the traditional song, The Twelve Days of Christmas, to complete the table shown below.

Show how you could represent the gifts given each day using simple pictures, a recursive formula, and an explicit formula.

• How many gifts are given on the 12^th day?

• How many total gifts are given over all 12 days?

On the eleventh page of the test my teacher gave to me, eleven Lions playing.

A while back, WOOD TV8 meteorologist, Bill Steffen, made a claim that the Detroit Lions’ record might be related to the amount of snowfall we get in Grand Rapids.  I emailed him asking for more information and part of his response is shown at the right.

• The scatterplot below verifies that a positive correlation does seem to exist between the number of Lion wins and Grand Rapids snowfall (based on the data provided).  Draw in an approximate line of best fit and use it to predict the amount of snowfall this season if the Lions win six games this season.

• By Steffen’s own admission, he “got to pick the years”, which is a factor.  Describe another sampling method that might better represent the relationship between wins and snowfall.

On the tenth page of the test my teacher gave to me, ten snowmen chillin'.

The idea of using snowmen glyphs to represent some of our winter and holiday preferences was introduced here. Recall that if the person making the glyph wears a hat during a typical winter day, then he or she would fill in the snowman's top hat. The same goes for wearing a scarf. Given the 10 snowmen shown below, answer the following questions.

• What percent of the people represented by these glyphs wear hats on a typical winter day?
• What percent of the people represented by these glyphs wear scarves on a typical winter day?
• What percent of the people represented by these glyphs wear hats or scarves on a typical winter day?

On the ninth page of the test my teacher gave to me, nine cookies baking.

As you might suspect, Mrs. Claus is a wonderful baker, especially when it comes to cookies. Her specialty is called Santa’s Hats, which are basically isosceles triangles with red frosting and sliced marshmallows across the bottom and on top. Because so many of the cookies she bakes are for the elves, Mrs. Claus thinks of a batch of cookies in terms of three-quarters of a dozen. If each of these batches takes eleven-and-a-quarter marshmallows, then how many marshmallows are required for a dozen cookies?

On the eighth page of the test my teacher gave to me, eight reindeer playing.

You know Dasher and Dancer and Prancer and Vixen, Comet and Cupid and Donner and Blitzen: But do you recall… what you learned in math class at all?  Use these eight famous reindeer (sorry, no Rudolph) to solve the following problems.

• Only two reindeer can lead the team.  How many different leading pairs are there (left side and right side does not matter)?
• You have probably heard about the Reindeer Games. Did you know that they only give ribbons for 1^st, 2^nd, and 3^rd place? How many different ways are there for the three ribbons to be awarded the eight reindeer? (Assume that each reindeer is equally likely to win.)
• Which of the above questions is a permutation and which is a combination?  Please be sure to explain your rationale.

On the seventh page of the test my teacher gave to me, seven candles burning.

During the week of Kwanzaa, families gather in the evenings to light the candles of the kinara and discuss the Nguzo Saba or seven principles. There are seven candles – three red candles to the right, three green candles to the left, and one black candle in the center of the kinara.

A manufacturer of Kwanzaa candles did some research and found out that the candles do not all burn at the same rate. Below are box-plots representing the number of minutes it took for each candle to burn completely, separated by color (100 candles of each color were tested).

Which candle color seems to last the longest?  Describe how the box-plots support your reasoning.

On the sixth page of the test my teacher gave to me, six bulbs a blinking.

In order to save money this year, Dave decides to buy outdoor bulbs at Ben’s Bargain Bulb Bin.  The bulbs there are cheap, and there are a lot of them (some say an infinite amount), but only three out of every four bulbs work. Dave needs 6 bulbs for the end of each snowflake in a string of lights, but if even one bulb doesn't work, then the entire snowflake remains dark. What are the chances that a snowflake using bulbs from Ben's will be unlit?

On the fifth page of the test my teacher gave to me, five Pecan Puffs.

Just in time for the holidays, a new item has been added to the J. Peterman Candy Collection – Perfect Pecan Puffs. These are five pecan puffs packed in cube-shaped boxes that are arranged to look like a "P" (see ad below) held together using green holiday paper.

The paper is expensive, so Elaine suggests using a "net" to cover the packages. What might this net look like?

PS: Kramer thinks that this is the perfect gift for Festivus!

On the fourth page of the test my teacher gave to me, four lighted trees.

Clark Griswold just purchased The Growing Tree Kit to add to his Christmas light show. Each kit has four, green, foot-high triangles/trees and four strings of red lights that fit around the perimeter of each tree. What makes these trees special is that they stack to make larger trees. Then the string of lights can be put around the perimeter of the larger, combined tree (see the examples below).

Single tree       Combined tree

When Rusty puts together the combined tree, he notices that two of the strings of lights are left over. "That's not a problem," says Clark. "We'll just add them to the roof. What I'm worried about is the visibility of the tree. I think we need to make it bigger - maybe 10-feet tall."

In order to make a 10-foot tree that is mathematically similar to the smaller trees, how many kits does Clark need to buy? Also, how many strings of lights will be needed to light the 10-foot tree's perimeter? What do you notice about the area and perimeter of these similar triangles as they dilate?

On the third page of the test my teacher gave to me, three wise men.

A version of this problem can be found in
Connected Mathematics Project's
Clever Counting

In the story of the Magi, three wise men come from the east looking for the king of the Jews. On their way, they stop at King Herod's for directions. They find out that the Messiah was to be born in Bethlehem. Imagine that the network shown to the right represents the many paths that the wise men could have taken.

• How many paths were there from the East (E) to King Herod’s Temple (H)? Include only the shortest paths (south/down or west/left).
• How many paths led from King Herod’s Temple (H) to Bethlehem (B)? Include only the shortest paths.
• In all, how many shortest paths are there from the East to Bethlehem that pass through King Herod’s Temple? Explain how you arrived at your solution and why it works.

On the second page of the test my teacher gave to me, two types of stools.

From America Trek

In a shed at the North Pole are the parts for 75 stools for the elves in Santa's Workshop. Some of the stools will be three-legged and some will be four-legged (for the bigger elves like Buddy). The shed contains 75 seats and 259 legs. If Santa does not want any parts left over, then how many of each type of stool can be made with these parts? Please use two different representations (pictures, words, symbols, ...) to communicate your thinking.

On the first page of the test my teacher gave to me, a Dreidel with a counting tree.

The Dreidel is an integral part of the celebration of Hanukkah. Children play by spinning a Dreidel and depending on the side it lands on they win or loose candy. Using the modified rules shown below and this online Dreidel, investigate what happens on two successive turns.

Specifically:

• What are the chances that the combination of these two turns will result in a negative score?
• What is the mean number of points you earned as a result of spinning the Dreidel twice?

Conduct ten trials and record the cumulative points won or lost as a result.

Dave conducted 990 trials of his own (see the table below). Add his results to your own, and answer the questions of interest. Why is this experimental probability?

Points	40	25	20	18	10	5	3	0	-2	-4
Results	57	127	127	121	61	124	111	61	136	65

Check your results against the theoretical probability. Be sure to support your work with some sort of representation (organized list, tree diagram, table, area model, or network). Why is this theoretical probability?