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The What, the How and the Who (by Brad Fulton)

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When I first began teaching math, I focused on what math skills needed to be taught. It was a changing time in California mathematics with the old guard and their traditional ways pitted against us young, progressively-minded newcomers who wanted to have students explore numeracy and solve the problem-of-the-week. At the end of each day, I slumped at my desk and asked myself, “Now that I have taught that concept, what comes next?” The what of mathematics consumed my thinking. We were moving from a pencil-and-paper age to a calculator-and-computer age, so it was a time of redefining the content of mathematics. Whereas my elementary teachers simply turned to the next page of the textbook, we were asking, “Is it necessary to learn to master long division with decimals?
How important is factoring in this digital age?”

Over the years the pendulum of mathematics swung back and forth between focusing on conceptual understanding or on computation. During these swings, I began to explore the how of teaching. What is the best way to communicate mathematical thinking and processes to the student? Do we begin with a problem? with manipulatives? with a procedure? The emphasis of the how over the what redirected my teaching. The what was still the destination, but the how was the vehicle that would get us there.

This idea was confirmed when I recently read Chris Shore’s blog on the “4½ Principles of Quality Math Instruction”. In the article, Shore noted that the top performing mathematics nations do not demonstrate a similarity in their instructional techniques. However, they do have common underlying principles. Although the what is taught in various ways from one high performing nation to the next, those nations share certain principles. These were the same principles that govern successful math teachers everywhere: Commitment to a few high standards, initial instruction in concepts prior to procedures, good questioning strategies and high accountability.

The focus on how I taught mathematics occupied much of my planning time. However, as the years wore on, I found that I was tiring of the profession. I was floundering through my fourteenth year in education when I learned the lesson that changed my teaching forever. Misbehaving students were making every day a nightmare. I was ready to quit when one morning I decided I had had enough. No longer was someone else going to determine what kind of a day I was going to have. From now on I was going to have a good day regardless of what the students tried to do. I started smiling at their snarling faces. I talked kindly even when they were surly. I commented on their new shoes, asked about their interests, and commended their
accomplishments. The transformation produced immediate results as I connected with my students for the first time.

That’s when I learned that who I taught mattered much more than what I taught. In the past, I had always tried to learn more about math. Now I tried to learn more about my math students. I discovered that students don’t learn math from mathematicians; they learn from people who care about them, and for the first time, they had my heart as well as my head.

There is a wonderful children’s book called The Velveteen Rabbit in which a brand new stuffed toy is given to a boy. Through the years, the boy loves the rabbit and holds it so closely that eventually all its fur is rubbed off. When he grows, the old rabbit is tossed out in the yard. That night the rabbit awakens for the first time to find that he has become real because he was loved. I thought I was a teacher when I first held my brand new teaching credential and state-adopted textbook. Over the years, my students pulled me so close to their hearts that all my hair has since been rubbed off! They grew up, moved on, graduated, and left me behind, but in the process they loved me and I loved them, and somewhere along the way, I became real. We can’t teach the what until we have answered the how. And we can’t teach at all until we know the who.

Brad Fulton, teaches in Redding, CA. He runs Teacher to Teacher Press, with Bill Lombard. The materials and lessons they offer stress conceptual understanding and problem-solving in very creative ways. Brad is an experienced and engaging presenter. I had the chance to sit down to lunch with him at the CMC-South conference this month. We were two old dogs talking about how “we have been doing common core long before they called it common core.” The conversation of how math education has progressed over the last two decades and about the “half principle” led me to invite Brad to write this guest blog.
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Multiple or Best Reps?

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I find it interesting that on the day that we post our most recent lesson, 4 x 4, (sample page), Dan Meyer posts the question: Aren’t we doing kids a disservice by emphasizing “multiple representations” rather than the “best representations?”

I understand where Dan is coming from…Why the overkill, when one proper tool solves the problem. I have three quick responses to this.

1) If the goal of the current activity is to apply previously learned skills, then I agree with Mr. Meyer. Students should develop the savvy to choose the most appropriate tool at hand, and implement it properly. When faced with embedding a nail, is there any sense in using both a hammer and a rock?

2) If the goal of the lesson is to build conceptual understanding of the four formal representations of a linear relationship (words, equations, data, graphs), then generating the other three from any given representation develops this insight. How many students can graph a line by plotting the y-intercept and then counting the slope up and over, but have no idea that they just stated the infinite set of points that satisfy the equation?

3) If the goal of the day is to offer a point of access to the students, then the temporary representation will eventually give way to a higher level of abstraction. Look at the banner on Christopher Danielson’s blog. These multi-link cube models can represent the various ways to factor the number 24. Alongside these 3-dimensional arrays, students could be representing factors symbolically, 2 x 2 x 6, 2 x 3 x 4,  3 x 8 etc. In time, the blocks are left behind for a level of abstraction that is far more efficient. Afterall, it is faster to write the factors on paper than it is to build them with the blocks, especially when the students start factoring much larger numbers. So to push back a little bit on the original question: Are we doing kids a disservice by offering training wheels when learning to ride a bike?

The answer to the question of “multiple representations” or “best representation” is, as always, up to the judgement of the teacher at the time.

Blog, Common Core

7 Lessons Learned about the Common Core

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Here is my list of lessons learned at The California Math Council’s South Conference, in Palm Springs. The overall theme of the conference was implementing the Common Core State Standards.

1) The new standards are truly world-class. I wish my own children went through school in the common core era. The expectations of the next generation of students are far higher than anything the state expected of my son and daughter. If our nation can rise to this new bar, we will finally be on par with the top performing countries around the world.
Message for teachers: If we only meet a small fraction of these expectations, the students of the future will still receive a better education than those sitting in our classrooms now.

2) Leaders are optimistic about the train wreck that is about to happen. There is no way that teachers will get students ready in time for the first wave of common core assessments. The change is too great, too quickly. The initial results promise to be abysmal. The question is from that point whether we will rise from the wreckage and move forward, or back pedal towards the old ways. Dr. Bill Schmidt and Tim Kanold are both optimistic that states, schools and teachers will continue to train, learn and eventually meet expectations.
Message for teachers: Brace for impact, then take advantage of the special opportunity that is being presented to us.

3) “We have been doing the common core long before there was a common core.” I heard this phrase from Brad Fulton and several of the reform leaders and innovators at the conference. The Math Projects Journal could make the same claim. For the 14 years of our publication, we have pushed for limited topics, conceptual understanding, higher order thinking and holding students accountable to all the above (4.5 Principles). This is not because anyone of us had unique ideas on education. We were just following the international research studies. I have heard that education is always 20 years behind the research. Any coincidence that California with conduct its first Common Core assessment 18 years after the publication of the TIMSS report?
Message for teachers: Educate yourself; what seems new really is old school.

4) The answers are out there. There was plenty of information on instruction, assessment and professional development. There are no secrets on how to implement the new standards. The only real question is how quickly and pervasively will that information find its way into classrooms. Tim Kanold claims, with evidence, that the best vehicle to deliver the information and practices is teacher collaboration.
Message for teachers: Have your school adopt a PLC model.

5) The Common Core emphasizes teaching practices as much as content. I was aware of the practices listed in the Common Core documentation, and though I regularly implement most (but admittedly not all), they always seemed more like guidelines than rules. (Problem-Solving, Reasoning, Modeling, Arguing, Tools, Precision, Structure, Patterning) In all the featured presentations, there was the consistent message that students learn as much from how we teach as from what we teach. Dr. William Schmidt of the University of Michigan claims that our current practices lack the logic and structure that is inherent in our subject matter.
Message to teachers: You will not only change what you teach, you will change how you teach.

6) One-third of the content of most textbooks can be thrown out. Dr. Schmidt led a study in which teachers corresponded their lessons and accompanying textbook pages to the Common Core standards. The study discovered that on average, one-third of the textbook content was avoided. There really will be time to slow down and to teach problem-solving.
Question for teachers: How well are you going to use the extra three months?

7) “Technology needs to mean more than paper on an iPad.” Dan Meyer gave a compelling presentation on the use of technology to push students to higher levels of thinking. He said that currently many teachers and companies are simply moving textbooks and lesson plans over to electronic devices as scanned material. Much of Mr. Meyer’s presentation was on the unique ways in which videos and photos can be used to perplex students.
Message for teachers: Times are a changin’. It’s time to catch up.

For the alarming number of those who had not heard much of the Common Core, it was a terrifying weekend. For those of us who have embraced its values over the last two decades, it was an exhilarating, yet still disconcerting, conference. Big kudos to the committee for putting on such a hugely successful event. Thank you for helping point us all in the right direction.

P.S. Catch ’em 2012 from Chris Shore’s presentation at CMC.

All Subjects, Blog

The Elusive Relationship of x & y

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“Ali has $10 dollars and spends $5 dollars every 2 days.” That was the simple scenario offered for the linear relationship that my students were working with at the time. I was having them write equations from various contexts in order to emphasize the concepts of slope and y-intercept. Boy, did I unravel a serious issue.

Typically, what is asked when dealing with slope-intercept form of lines is somemthing like: “What is the slope and y-intercept of the following equation:  y = -(5/2)x + 10?” Then those values are to be used to graph the equation. The hope commonly is that the students have been paying enough attention to know that the “slope is the number in front of the x.” The frustration commonly is that students still can’t identify the slope from the equation. From a context, the slope is even more difficult to identify, like with the Ali Scenario above. What I found out through a very unique question is why it is so difficult for them.

In the previous contexts that I had offered them, the students would see only two numbers. “Sally has 1 friend and makes 2 new friends every week,” as an example. I was asking them to identify the constant and the rate of change in examples like this. That was a struggle, but they eventually started to get it. When they ran across the Ali Scenaro, I was expecting the three numbers that were representing only two values to cause issues. I was correct. But it was their response to my next question that opened my eyes: “What two quantities are related here?”

I got blank stares. So I asked them to write down the two words that represent the quantities being related in the given context. When I looked at their answers, I was shocked to find the two words that over half the class had written down: spend & has!

Really?! I wrote those two words on the board, and asked them, “Does it make sense that the number of spends that I have determines the number of ‘hasses’ that I have?” I noticed everyone’s attention in class was riveted on the two words on the board. They knew what I was saying didn’t make sense, but they were struggling to reconcile the issue, so I asked another question. “What determines what in this scenario?’ A student finally offered, “days determines dollars.” Yes!

We had to wrap up the day, since I had spent so much of the period mining this one context for mathematical understanding. So the next day, I started the lesson by posting the same Ali Scenario and the two sets of words, “spends & has” and “days & dollars.” We revisited the idea that even though the sentence claims that “Ali spends and Ali has,” we are looking for words that represent numbers. Then we can use those numbers and a few symbols (variables, operations and equal sign), to write an equation (an abstract generalization) to represent the relationship. That equation (like the scenario and eventually the graph) represents an infinite number of combinations of “days & dollars.”

The students now easily identified 10 as the constant and wrote it properly as the y-intercept of the equation. The trouble was in dealing with a slope that was represented by a negative fraction, which was the issue that I originally intended the scenario to pose. However, my adventure into the minds of my Algebra students helped remind me that we too often have them leap from the concrete to the abstract, or that we skip the concrete altogether. How many of the millions of Algebra students in this country are graphing linear equations, yet have no idea that x and y actually represent a quantitative relationship? Focusing on getting the answer correct (“Graph this by starting at this number and counting up and over this number.”) often times bypasses the ultimate goal: mathematical understanding.

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“Too Bad We Can’t Find Out Which Way Works Best.”

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“It’s too bad someone can’t do a study to see which way (direct instruction or hands-on learning) works best.”
This comment came from a colleague in a discussion on how well my remedial Algebra class did with the assessment on a particular non-traditional lesson. Before I share my thoughts on this, let me offer some background on the class and some data.

The demographics of the class is quite challenging: 4 SDC (special day/workshop), 5 ELL (english language learners), 7 IS (special ed), 5 Academy (highly at-risk), out of a total of 30. While the others don’t have an acronym after their name, they still have a history of struggling in school. And I absolutely love teaching this group! I share all this first, because the data I am about to show will be all that much more impressive.

I began our previous unit on Solving Equations with a pre-assessment of the 6 types of equations that they were going to learn to solve:

1) x + 4 = 31
2) 4x = 28
3) 7x + 5 = 26
4) 10x + 2 – 4x = 44
5) 11x – 4 = 3x + 12
6) 9(x – 2) = 45

Then I spent a week of direct instruction (D.I.) and 2 days with the simplifying and linear equations components of the Truffles lesson, after which I assessed them for a second time on the same 6 equations. I then led the students through the Hippity-Hoppity lesson for 4 days, and assessed them again on the same 6 equations. The progression of results is shown below. (There are only 22 students shown due to the shuffling of students classes at the beginning of the year.)

 # correct Pre-
Assess		 After
Truffles & D.I.		 After
Hippity-Hoppity
6s 4 13 19
5s 4 7 3
4s 3 1 0
3s 3 1 0
2s 3 0 0
1s 2 0 0
0s 3 0 0

My class went from only 36% getting 5 or 6 correct, to 90% after the Direct Instruction and Truffles lessons, to 100% after the Hippity-Hoppity lesson. It is worth noting that the number of students correctly solving all 6 equations rose from 59% to 86% after the last “active-learning” activity.

My friend made his “too bad we can’t find out which way is best” comment having only heard about the last lesson, not knowing the work I put into the unit throughout. That work demonstrates a variety of strategies that might be classified as direct instruction or hands-on learning. He didn’t realize that I had implemented several ways, not just one.

I utilized several strategies because there is a great deal of research that shows that the best way is a balanced approached. However, that balanced approach is not between methodologies (direct instruction vs discovery/hands-on/active learning); it is between skill acquisition and critical thinking. I chose to use lectures and guided practice to impart skills, the Truffles lesson to instill understanding of a variable, and the Hopping lesson to offer an application of the topic. These last two lessons also required my students to practice other skills that they lack: reading and writing in a mathematical context, and following multi-step directions.

So we should tell all of our colleagues that we do indeed know a best way. It is not my way, your way, or their way, but a balanced way.

All Subjects, Blog, Boot Camp/Number Sense

Number Tricks (student sample)

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Number Tricks is a lesson that involves writing and simplifying expressions. It demands the higher order thinking skills called for in the Common Core in several ways. 1) The students are to write a mathematical model for a trick given to them. 2) They are to create their own trick and offer the algebraic expression that represents it. 3) It presses the students to understand the concept of a variable; in this case, the variable represents the number originally chosen. 4) The students are asked to compare their simplified expression to the pattern generated by the various numbers tested. The lesson offers a great opportunity for a high level of critical thinking with a rather low level piece of content.

Here is an erroneous submission from my Algebra class. I want to analyze the mistake and discuss why this lesson was so very good for this student even though the “answer was wrong.”

This was Dewey’s response to creating his own Number Trick, including 3 numbers to generate the pattern, and the algebraic expression it represents:

Pick a number 3 10 -7 x
Add 4 7 14 -3 x + 4
Multiply by 2 14 28 -6 2x + 4
Subtract 3 11 25 -9 2x + 4 – 3
Subtract the
Original Number		 8 15 -2 2x + 4 – 3 – x
Simplified: x + 1
Common Result: one more than the number picked

Now of course we can see that the student should have included the parenthesis when multiplying by 2. The final expression should have been:

2(x + 4) – 3 – x, simplified: x + 5

So the positives? The student is showing that he is solid in his operations with negative integers, that he can simplify correctly and that he is interpreting the final expression properly (x + 1 means 1 more than the original number). The Big Negative? The pattern of numbers does not support the students simplified expression. The resultant numbers are NOT one more than the original number; they are 5 more.

My judgement call here was to ask Dewey if his expression matches the pattern. He couldn’t answer right away. There was disconnect between generating the expression and actually knowing what the expression represented. Once I pointed out that the last numbers in each column where not 1 more than the first, I asked him to find his own mistake, which he did. Dewey was then able to correctly simplify and without assistance verify that his new expressions supported the pattern of numbers.

Dewey did a great deal of complex thinking with a topic as simple as simplifying expressions.

All Subjects, Blog

Standard & Slope-Intercept Forms Don’t Play Nice

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We have contexts for equations in slope-intercept form:

Johnny has 4 friends and makes 3 new friends every 2 weeks.
             f = 3/2w + 4

We have contexts for equations in standard form:

Adult tickets cost $5 each and student tickets cost $3 each, for a total of $150.
                                  5a + 3s = 150

And we ask students to convert from one form to another.

                f = 3/2w + 4        to        3w – 2f = -8
                                             and
                5a + 3s = 150       to        s = –5/3 a + 50

But you know what doesn’t convert very well? … the context from one form to another.

Take Johnny’s Friends for example. The 3/2 means 3 friends every 2 weeks, and the constant represents the 4 friends that he started with. After converting to standard form, what do the coefficients 3 and -2 represent? What does the constant -2 represent? Maybe the context is there, but a typical algebra student just isn’t going to take the time find it, or maybe the context was lost in the conversion. The key to the question (but not to any viable answer) is in the units. If we apply some quasi dimensional analysis here:

                  f = 3/2w + 4
                 friends = (friends/week)weeks +  friends
                 friends = friends +  friends
                friends = friends

Check. Now try our quasi dimensional analysis on the standard form conversion: 3w – 2f = -8. Say what? Exactly! We can do it with our Ticket equation, though:        

                  5a + 3s = 150
                 (dollars/ticket)tickets + (dollars/ticket)tickets = dollars
                 dollars +  dollars = dollars
                dollars = dollars

Check again. And the context here actually converts somewhat. We can say that the equation s = –5/3a + 50 tells us that the number of student tickets is equivalent to fifty total tickets minus five-thirds of every adult ticket. If you can hang with that, try it on the following context.

Buddy sells cupcakes for $2 dollars each and brownies for $1 each, for a total of $14.
2c + b = 14            

When you convert  this equations to b = -2c + 14, the 14 miraculously changes from representing dollars to representing number of brownies.

While the context might get lost in the translation between these forms of an equation, I know from my mathematics experience that there is still usefulness in being able to move fluidly from one form to another. In fact, to a mathematician, context is often burdensome. The beauty of naked math problems is that their abstraction transcends an infinite number of contexts. I just don’t think that is where you want the typical algebra student to start.                

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Math Saves ASB Election

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Last spring, I was walking through the front office of school when I was pulled into the ASB office. The director and his secretary appeared fearfully perplexed. They had the results of the election for ASB President, but could not determine a winner, so they needed the help of a math teacher. The winner was scheduled to be announced in less than an hour. Here was there quandry:

They advertised that the election would be based 20% on the Interview, 20% on Teacher Evaluations and 60% on Student Voting. Those results stared at them from a white board on the wall.

   Interview              Evaluations           Voting
Candidate #1       Candidate #1       Candidate #3

However, they could not determine from this data who won the election. I asked if they they had all standings in all three categories.They affirmed that they did and added those results to the whiteboard.

   Interview             Evaluations           Voting
Candidate #1       Candidate #1       Candidate #3
Candidate #2       Candidate #2       Candidate #1
Candidate #3       Candidate #3       Candidate #2

With this information, I posed to weight the election categories much the same way that I do my gradebook. Each candidate’s score was the sum of the three products of the Candidate’s place in the category (3 being highest score, 1 being lowest) and the percentage given that category. Thus,

#1 = 3(0.2) + 3(0.2) + 2(0.6) = 2.4
#2 = 2(0.2) + 2(0.2) + 1(0.6) = 1.4
#3 = 1(0.2) + 1(0.2) + 3(0.6) = 2.2

Candidate #1 wins!

I know there are several other mathematical methods to determine the winner. I invite those solutions in the comments below. My purpose for sharing this antecdote is to pose the question. “Why did it take a math teacher to figure this out?” These were intelligent, educated people struggling with this problem, “Why did they need a mathematician to resolve it?” The math involved is not high level, nor is this type of problem in any math curriculum that I experienced, so “How was I more adept at solving it than they were?”

I am going to surmise (inviting rebuttal), that it is my experience in solving math problems, not my content knowledge of math, that led me to the solution. I definitely have taken more math courses than the Activities Director, yet I did not need Trigonometry or Calculus to crown a champion. I needed simple problem solving skils in a mathematical context. So The Election Problem spurs me to pose more “problems” and fewer “notes and exercises” to my students.

You?

Blog, Boot Camp/Number Sense

Teaching Order of Ops with the 4-Digit Problem

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I teach a low-level Algebra class (Algebra Essentials). Half of the thirty students are Special Ed or English Language Learners. I was teaching Order of Operations, and rather than offer notes on a subject that they had seen several times before (and still failed), I taught the topic with the 4-Digit Problem instead. (created by College Prepetory Mathematic program, UC Davis)

The 4-Digit Problem goes something like this: You must use four of the same digit (I use 8) to form expressions that have designated values. You may use any of the standard operations represented in “PEMDAS;” you may combine eights to make 88; and you may use any number other than 8 only as an exponent (which then makes roots legal). I did offer some hints:
8 ÷ 8 = 1          80 = 1          √(8 + 8) = 4          3√8 = 2.

I had the students practice by first attempting to express the value 19 with four 8’s. I then had them independently attempt to express the values 1-5. Here is what this supposedly low-level group came up. (multiple solutions shown)

1 =  8 ÷ 8 · 8 ÷ 8     or     8 – 8 + 8 ÷ 8

2 = 880 + 880      or     8 ÷ 8 + 8 – 8

3 = 8 ÷ 8 + 80 + 80     or     80 + 8 ÷ 8 + 80

4 = 80 + 80 + 80 + 80     or     8 ÷ 8 · √(8 + 8)

5 = 3√8 + 3√8 + 3√8 – 80

Their class/homework that day was to then generate the values 6-10, and submit for a grade.

Great day for learning Algebra!

All Subjects, Blog

First Day Challenge (cont’d) – The 6 C’s

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Hopefully, my challenge pressed you to create a new and unique First Day routine. As I shared in my previous post, I was inspired by the story of the martial art student who claimed that for each good teacher he had in China, all the principles that he would master with them were taught in the very first lesson. So I came up with the 6 C’s of the Math Mission. These are basically my answer to the question “Why should we learn this?”

    1. Conceptual Understanding of mathematical principles and demonstration of Procedural Fluency,
    2. Critical Thinking in a mathematical context,
    3. Communicating Reasoning in a technical field,
    4. Constructing Models of the natural world,
    5. Creativity expressed freely and joyfully in the problem solving process,
    6. Collaboration with others.

At the beginning of my first day with the students, I have them take a  4-question Opening Quiz. Each question is headed by and supports one of the first four of the 6 C’s above, which are the criteria for our state test. I have the students take the quiz with the explanation that it is like a movie trailer. Rather than a test on something I hoped they learned last year, the quiz is a  preview of coming attractions. Therefore, I don’t expect them to be able to answer all four questions; however, I do expect them to give an intelligent response. I preach to them that we cannot always give a complete response or even an accurate response, but we can always, always always give an intelligent response. Showing numbers, equations or diagrams is intelligent; leaving the paper blank or writing ‘idk’ is not.

The students work independently for a few minutes while I do the normal housekeeping duties, like checking their course schedules and adding students to the roster. Then I have them share ideas with each other. If they like what they see on someone else’s paper, they are free to write it on their own as long as the person explains it to them. “We share; we don’t copy.”

Then comes the time for a whole class discussion. I use this opportunity to inform the students that this process of independent thought, group sharing and class discourse  (think, pair, share) is a normal routine in the class. I point out that the 6 C’s are displayed on the front wall, because this is what I come to work to do each day (my goal). I am not there to get them a good grade (their goal). Of course, if I achieve my goal, they will achieve theirs. So I ask  for volunteers on each of the questions, stressing the appropriate meaning for each of the 6 C’s:

The second half of the period is committed to getting to know the students. We start with the first student who stands up, states their name and something interesting about themselves. The statement of interest is really just a stall tactic so that I can review the names that have already been mentioned. After every student has done this, I go back and state all their names. And finish the intros with some brief information about myself.

I close the class by telling them that the order of things today was intentional and symbolic. That the class will be about math first and foremost (not points and homework), and of course, it is each one of them to whom I must teach math, so they are important as well. I reiterate some of the points of the class culture like giving an intelligent response, think-pair-share, and “share, don’t copy.” I hand them the grading policy as they leave. We will discuss each of its points when the time comes. Hopefully, I communicate everything in a dynamic manner so the students anticipate a fun year full of learning.