(cross-posted from mathcoachblog.com)

At our recent spring conference, Jim Rubillo, Annie Fetter and I were saying our good-byes at the end of a fun evening, when Jim’s puzzly side emerged…

Every time I talk to Jim, he’s bound to have a neat problem for me to chew on. The last time, he shared a fun task involving the harmonic series. Take a few minutes and think about this Pascal’s Triangle scenario…I’ll even leave you some spoiler space.

At the ATMOPAV Spring Conference last month, Jim shared an entertaining talk titled “Gambling, Risk, Alcohol, Poisons and Manure – an Unfinished Life Story”. The talk led the attendees on a journey through the history of statistics, starting with games of chance and the meeting of Chevelier du Mere and Blaise Pascal, through the introduction of formal inference procedures developed at the Guinness brewery, and to identifying statistical abuses in the present day. Jim is a life-ling educator, the former Executive Director of NCTM, and a past ATMOPAV Presdident. It was a thrill having him share his ideas with the group, and we look forward to seeing him again at future ATMOPAV events! |

Had enough time to think about this Pascal question? Spoiler time is up! So, which rows are in Pascal’s Triangle are we talking about here? In theory, we are talking about “all” of the rows in the infinite Pascal’s Triangle, which makes this a bit tricky to think about for kids (and adults as well!). But Jim shared with me slides which show the proportion of evens in increasing numbers of rows of the triangle. You will notice that as the number of rows grows, the proportion of even entries also increases, and approaches 1. What a neat result! To the right is an animated gif I made using a Pascal’s coloring applet which shows the increase in the proportion of even (white space) numbers in increasing rows. For your class, this is a fun opportunity to talk about the parallels between Pascal’s Triangle, Sierpinski’s Gasket, and fractal area. |

By Michael Eiseman, "Algebra By Hand" Dear US teachers of algebra, I have a compliment, a question, and an answer for you. First the compliment: I like the way you explain algebra to your students by applying equivalent arithmetic operations to both sides of an equation as depicted in Animation 1. I think this is a great approach because it explains why algebra works and leads directly to a procedure in which your students may develop fluency.Do you know that Russia, China, India, Japan, or Malaysia use a different approach? They teach their students to treat terms and factors in an equation as "movable objects" as depicted in Animation 2. Algebra is simply introduced as a conceptually unfulfilling symbol manipulation exercise. Have you ever explained to someone from India how we teach algebra in the US? Their unanimous reaction is, "Why would you do it that way? It seems so complicated." Yes, it is more complicated, but it is better-connected to fundamentals.Why would Eastern countries take this approach? They do this because it reduces their students' cognitive load relative to the US approach. Again, my opinion here, I think they are missing a learning opportunity. HOWEVER, (and the question is coming) dig inside the mind of any practicing engineer, scientist, accountant, or analyst in the US who uses algebra often, and you will invariably find that they use the shorter, Eastern country method of moving terms and factors and NOT the fundamentally-based procedure you taught them. Question:And finally, the question: "Why do all your best students eventually find that they must cast aside the procedures you taught them?" Answer: If you can picture the look of horror on their college calculus professor's face when the student writes "-8" on each side of an equation before progressing to the next solving step of a long algebraic proof, you already know the answer. That calculus professor expects your student to have progressed past applying equivalent arithmetic operations to both sides of an equation. That calculus professor expects your student to have what I call Advanced Procedural Fluency (APF). APF is the ability to quickly solve linear equations by moving terms and factors in your head as taught in eastern countries. As an algebra teacher, it is highly likely you possess APF. Think about WHY you developed this skill on your own. You invented it because you could see that it was faster and more accurate than the method you teach your students. You realized that success in higher math requires APF - that solving by applying equivalent arithmetic operations is too cognitively taxing, too much of a hindrance to permit advanced mathematical exploration. But if a US student does not eventually develop APF, they are mathematically handicapped relative to their counterparts in Eastern countries who will be able to solve linear equations more quickly and accurately. AFTER students have mastered conceptual understanding, procedural fluency, and application, we need to teach APF. If we don't, they will be forced to synthesize it on their own. Not only would teaching APF more quickly advance students who wish to pursue STEM-related careers, it may make algebra attainable to students who would otherwise find it incomprehensible using today's pedagogy. What do you think? Leave a comment and let me know. |

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By Marylouise Finch, Educational Consultant

This is a timed activity, with small groups of students creating math vocabulary graphic organizers. They will pass it around in their group and take turns filling out different corners until it is complete. The graphic organizer is the standard four-part model, with the vocabulary word in the middle: one corner is an example, one corner is an explanation, one corner is what it is NOT, and one corner is a picture. This can be adapted however you like.

-- Students form groups of four or less.

-- Each group is given a list of review math vocabulary words and a stack of letter-size paper (I often used scrap paper that had something printed on the other side).

-- Set the time for (about) 5 minutes.

Instructions:

1. Each student takes a piece of paper and folds it into quarters to divide into four parts and then unfolds it. Each student picks a different word from the list and writes the word in the middle of their paper, make sure to cross out the word they pick on the list.

2. When the timer is started, each student completes any one corner of their graphic organizer that they choose during the five minutes.

3. When the bell rings, they pass it to the left and the next person fills in a different corner.

4. When a paper is complete, they put it in their pile and start a new one.

It can be done for as long as you want, and also be a competition to see which group fills in the most. I grade them according to how accurate and thorough they were done, and also their conduct as I observed them.

]]>This is a timed activity, with small groups of students creating math vocabulary graphic organizers. They will pass it around in their group and take turns filling out different corners until it is complete. The graphic organizer is the standard four-part model, with the vocabulary word in the middle: one corner is an example, one corner is an explanation, one corner is what it is NOT, and one corner is a picture. This can be adapted however you like.

-- Students form groups of four or less.

-- Each group is given a list of review math vocabulary words and a stack of letter-size paper (I often used scrap paper that had something printed on the other side).

-- Set the time for (about) 5 minutes.

Instructions:

1. Each student takes a piece of paper and folds it into quarters to divide into four parts and then unfolds it. Each student picks a different word from the list and writes the word in the middle of their paper, make sure to cross out the word they pick on the list.

2. When the timer is started, each student completes any one corner of their graphic organizer that they choose during the five minutes.

3. When the bell rings, they pass it to the left and the next person fills in a different corner.

4. When a paper is complete, they put it in their pile and start a new one.

It can be done for as long as you want, and also be a competition to see which group fills in the most. I grade them according to how accurate and thorough they were done, and also their conduct as I observed them.

Adapted from Introduction to Statistical Investigations, AP Version, by Tintle, Chance, Cobb, Rossman, Roy, Swanson and VanderStoep

Bob Lochel, Hatboro-Horsham High School

Before the Thanksgiving break, I started the sampling chapter in AP Statistics. This is a unit filled with new vocabulary and many, many class activities. To get students thinking about random sampling, I have used the "famous" Random Rectangles activity (Google it...you'll find it) and it's cousin - Jelly Blubbers. These activities are effective in causing students to think about the importance of choosing a random sample from a population, and considering communication of procedures. But a new activity I first heard about at a summer session on simulation-based inference, and later explained by Ruth Carver at a recent PASTA meeting, has added some welcome wrinkles to this unit. The unit uses the one-variable sampling applet from the Rossman-Chance applet collection, and is ideal for 1-1 classrooms, or even students working in tech teams. Also, Beth Chance is wonderful...and you should all know that!

In my classroom notes, students first encounter the "sky", which has been broken into 100 squares. To start, teams work to define procedures for selecting a random sample of 10 squares, using both the "hat" (non-technology) method, and a method using technology (usually a graphing calculator). Before we draw the samples however, I want students to think about the population - specifically, will a random sample do a "good job" with providing estimates? Groups were asked to discuss what they notice about the sky. My classes immediately sensed something worth noting:

"There are some squares where there are many stars (we end up calling these "dense" squares) and some where there are not so many."

Before we even drew our first sample, we are talking about the need to consider both dense and non-dense areas in our sample, and the possibility that our sample will overestimate or underestimate the population, even in random sampling. There's a lot of stats goodness in all of this, and the conversation felt natural and accessible to the students.

Bob Lochel, Hatboro-Horsham High School

Before the Thanksgiving break, I started the sampling chapter in AP Statistics. This is a unit filled with new vocabulary and many, many class activities. To get students thinking about random sampling, I have used the "famous" Random Rectangles activity (Google it...you'll find it) and it's cousin - Jelly Blubbers. These activities are effective in causing students to think about the importance of choosing a random sample from a population, and considering communication of procedures. But a new activity I first heard about at a summer session on simulation-based inference, and later explained by Ruth Carver at a recent PASTA meeting, has added some welcome wrinkles to this unit. The unit uses the one-variable sampling applet from the Rossman-Chance applet collection, and is ideal for 1-1 classrooms, or even students working in tech teams. Also, Beth Chance is wonderful...and you should all know that!

In my classroom notes, students first encounter the "sky", which has been broken into 100 squares. To start, teams work to define procedures for selecting a random sample of 10 squares, using both the "hat" (non-technology) method, and a method using technology (usually a graphing calculator). Before we draw the samples however, I want students to think about the population - specifically, will a random sample do a "good job" with providing estimates? Groups were asked to discuss what they notice about the sky. My classes immediately sensed something worth noting:

"There are some squares where there are many stars (we end up calling these "dense" squares) and some where there are not so many."

Before we even drew our first sample, we are talking about the need to consider both dense and non-dense areas in our sample, and the possibility that our sample will overestimate or underestimate the population, even in random sampling. There's a lot of stats goodness in all of this, and the conversation felt natural and accessible to the students.

Students then used their technology-based procedure to actually draw a random sample of 10 squares, marking off the squares. But counting the actual stars is not reasonable, given their quantity - so it's Beth Chance to the rescue! Make sure you click the "stars" population to get started. Beth has provided the number of stars in each square, and information regarding density, row and column to think about later. But before we start clicking blindly, let's describe that population. The class quickly agrees that we have a skewed-right distribution, and take note of the population mean - we'll need it to discuss bias later. |

Click "show sampling options" on the top of the screen and we can now simulate random samples. First, students each drew a sample of size 10 - the bottom of the screen shows the sample, summary statistics, and a visual of the 10 squares chosen from the population.

Groups were asked to look at their sample means, share them with neighbors, and think about how close these samples generally come to hitting their target. Find a neighbor where few "dense" area were selected , or where many "dense" squares made the cut, how much confidence do we have in using this procedure to estimate the population mean?

Eventually I unleashed the sampling power of the applet and let students draw more and more samples. And while a formal discussion of sampling distributions is a few chapters away, we can make observations about the distributions of these sample means.

Eventually I unleashed the sampling power of the applet and let students draw more and more samples. And while a formal discussion of sampling distributions is a few chapters away, we can make observations about the distributions of these sample means.

And I knew the discussion was heading in the right direction when a student observed:

"Hey, the population is definitely skewed, but the means are approximately normal. That's odd..."

Yep, it sure is...and more seeds have been planted for later sampling distribution discussions. But what about those dense and non-dense areas the students noticed earlier? Sure, our random samples seem to provide an unbiased estimator of the population mean, but can we do better? This is where Beth's applet is so wonderful, and where this activity separates itself from Random Rectangles. On the top of the applet, we can stratify our sample by density, ensuring that an appropriate ratio of dense / non-dense areas (here, 20%) is maintained in the sample. The applet then uses color to make this distinction clear: here, green dots represent dense-area squares.

"Hey, the population is definitely skewed, but the means are approximately normal. That's odd..."

Yep, it sure is...and more seeds have been planted for later sampling distribution discussions. But what about those dense and non-dense areas the students noticed earlier? Sure, our random samples seem to provide an unbiased estimator of the population mean, but can we do better? This is where Beth's applet is so wonderful, and where this activity separates itself from Random Rectangles. On the top of the applet, we can stratify our sample by density, ensuring that an appropriate ratio of dense / non-dense areas (here, 20%) is maintained in the sample. The applet then uses color to make this distinction clear: here, green dots represent dense-area squares.

Finally, note the reduced variability in the distribution from stratified samples, as opposed to random samples. The payoff is here!

Later, we will look at samples stratified by row and/or column. And cluster samples by row or column will also make an appearance. There's so much to talk about with this one activity, and I appreciate Ruth and Beth for sharing!

]]>Later, we will look at samples stratified by row and/or column. And cluster samples by row or column will also make an appearance. There's so much to talk about with this one activity, and I appreciate Ruth and Beth for sharing!