Category Archives: All Subjects

Dr. Jon Star Speaks HOT Heresy

May 30, 2014 Chris Shore 1 Comment

Dr. Jon Star, of Harvard University, gave a mathematically blasphemous speech at the 21st Century Mathematics Conference in Stockholm, Sweden last year. The presentation was titled, Neuroscience and Cognitive Psychology of Mathematics. His heretical statement was that mathematics does not teach higher order thinking skills; only the teaching of problem solving actually teaches problem solving. The Math Projects Journal has always preached the teaching of mathematical substance, or what is now commonly known as higher order thinking skills (HOTS), so we reached out to Dr. Star regarding his research.

The belief that just by learning math one gets critical thinking skills is also not well-supported by evidence.

MPJ
You must know that your claim stating that math does not inherently teach critical thinking is very unnerving to the math education community.

Dr. Star
Just to be clear about my goals in the Stockholm talk, I was trying to argue the following:

First, the belief that math plays some sort of special and relatively unique role (as a discipline) in promoting what you refer to as HOTS (Higher Order Thinking Skills) is not well-supported by evidence.

Second and related, the belief that just by learning/understanding math, one gets critical thinking skills as well (e.g., two for the price of one, without explicit or even implicit attention to developing HOTS), is also not well-supported by evidence. Certainly in some instances this does happen, but it does not appear to happen in any widespread way for ‘typical’ students.

And third, given that we do want students to develop HOTS, rather than expecting/hoping that these just emerge as a natural by-product of learning/understanding math, it is essential that we think about how to explicitly promote critical thinking and problem-solving in what we teach and how we teach math. With respect to this last point, arguably generations of math curriculum and pedagogy reformers have sought this same goal – teaching math such that higher order thinking skills develop. But evidence and intuition suggests that this is very hard to do. But certainly we should continue trying…

MPJ
Is it math, per se, that does not impart the HOTS, or is it the way we teach math that is inept in imparting these skills?

Dr. Star
I would say that both content and pedagogy are important, but it seems that pedagogy plays an especially important role. If we want students to be able to transfer knowledge to domains outside of math class – apply reasoning skills that worked in math class to other kinds of problems – it seems necessary to teach with such transfer goals in mind. There are many different (at times competing) pedagogical visions for how to teach math such that this kind of transfer is possible. Some feel that the best approach is to engage students in certain kinds of reasoning and communication that are believed to facilitate application of knowledge to novel situations, and others feel that a certain amount of practice in applying concepts and skills is necessary for future transfer. I can see potential merit in both of these approaches, although empirically there isn’t a lot of good evidence to point us in the right direction.

I would say that both content and pedagogy are important, but it seems that pedagogy plays an especially important role.

MPJ
The 8 Common Core Standards of Practice imply that habits of mind can be taught. In your view, do these practices have value?

Dr. Star
I think that the Common Core practice standards are admirable goals. However, as noted above, I think we are still struggling to determine the best ways to achieve these goals pedagogically.

MPJ
Anecdotally, educated people think, communicate and behave differently than uneducated people. I believe research bears this out as well. Is this then simply a non-associated correlation (people who already have educated traits get an education), or does a quality education truly transform an individual?

Dr. Star
Certainly some people do develop problem solving skills merely by learning math. Some of these people developed (or would have developed) both math understanding and HOTS even if they didn’t have a classroom or a teacher – they could have done so by themselves on a desert island, so to speak. Most people, though, definitely need math training to learn math content, and they need explicit instruction in critical thinking to develop higher order skills as well.

Most people, though, definitely need math training to learn math content, and they need explicit instruction in critical thinking to develop higher order skills as well.

MPJ
What advice do you have then for classroom teachers in the quest for teaching higher order thinking skills?

Dr. Star
Try to identify the places in your lessons where you hope students are developing higher order thinking skills, and consider ways that you can be more deliberate and explicit in your pursuit of and assessment of these goals. For example, ask your students about any broader connections they are making from the mathematical content of the lesson. Give students opportunities to apply what they have learned in a lesson to other mathematical and non-mathematical topics. Let students know what you mean by phrases such as “critical thinking”, “problem-solving”, and “logical thinking”; give students examples of what these practices look like as well as tasks that allow them to develop and experience these important competencies.

Let students know what you mean by phrases such as “critical thinking”, “problem-solving”, and “logical thinking”; give students examples of what these practices look like as well as tasks that allow them to develop and experience these important competencies.

**** Dr. Star may be reached at jon_star@harvard.edu
****For more of Jon Star’s thoughts on Math Education, see this Scholastic video on YouTube.

All Subjects, Blog

SMP Posters by MPJ

March 10, 2014 Chris Shore 16 Comments

I created my own posters for the Common Core Standards of Mathematical Practices. I combined the best from what I found from others and added my own structure. Necessity dictated my doing this for two reasons: 1) I wanted to respect others’ copyrights, and 2) I couldn’t find any that were appealing to secondary students.

With that said, I offer MPJ’s SMP Posters for use in the classroom. (For JPEGs, click images below.) Each poster here has the following features:

The summary of the Practice straight from the Common Core documents, as listed in that famous grey box

The verbage of the Practice written in kid-friendly, first person language

A single word that embodies the particular practice

A diagram that displays an application of the practice, using Algebra as an example so as to span both middle and high school

A group of words that relate

A list of questions that pertain

A clip art image of a high school student to drive home the point that the practices are for them and not the teacher

An instructive statement that includes the word “Think”

A special shout out goes to the Jordan School District’s SMP posters for elementary schools which were the initial inspiration for this set. Other sources include: Eastern Bristol High School and Carroll County.

All Subjects, Blog, The Paradigm

Kicking the Textbook Habit

February 23, 2014 Chris Shore Leave a comment

I have had several inquiries about an article I wrote many years ago titled, Textbook Free: Kicking the Habit. I am not surprised, because, in these days of Common Core roll-outs with few valid materials, teachers are having to create and find their own curricula. While the article is over a dozen years old, it could not be more timely, so I thought I would make it available again. I hope this helps encourage teachers that using textbooks as a resource instead of as scripture in the era of the New Curriculum can be easy and fun.

Textbook Free: Kicking the Habit

Originally printed in The Math Projects Journal in May 2001:

I kicked the habit! I am no longer a textbook junkie. I no longer rely on my daily fix of some publisher’s bloated curriculum. I am free of my addiction without the help of an arm patch, rehabilitation clinic or twelve-step program. I quit cold turkey. Here’s how.

At my school, the students are issued a math book that they leave at home and each teacher is issued a class set. I usually keep one underneath each desk. This year, however, the librarian informed me on the first day of school that we were out of Geometry textbooks. Our student population had grown so large that our library ran short. In fact, for two to three weeks many of my students would not have a book at home either. There was talk of teachers sharing class sets and photocopying pages for students. I decided to try a different strategy. I took this as a professional challenge to see how long I could teach without a textbook. I knew whatever happened would be a growing experience for me as well as my students.

Well, by no fault of the school library, two to three weeks stretched to seven. By that time, I was well into my “textbook free” strategy, so I just kept the ball rolling…for the rest of the year. I used only 12 assignments from the textbook in those 180 days. Here is how that unique experience of being textbook free has changed my teaching, forever.

Firstly, I am now much more focused on standards. Rather than leafing through the textbook, I looked at my state and district standards, and established my curriculum from those. After all, shouldn’t they be determining what we teach? From there, I grouped the topics into units, and then scheduled individual lessons. This process naturally pared down the number of topics that I taught and allowed me to allocate a full week of instruction to each concept, rather than one day to each section of the textbook.

The second big change that has occurred is the structure of my lessons. Everything from my homework to my instruction has radically changed. My typical textbook free lesson was comprised of three to six problems of various difficulty. Oftentimes, I began a lesson with one to three review problems from previously learned material which applied to the current lesson. This is similar to a traditional warm-up with the exceptions that the problems are very relevant to the new lesson, and not simply arbitrary review.

Sometimes, I began with THE big problem from the previous night’s assignment, and solicited student responses. It is not hard to see that my old practice of dedicating 20 minutes of class time to questions on how to complete the previous homework disappeared. The intent of the class slowly evolved from getting the answers correct to understanding the mathematical principles behind the question.

These introductory problems served as a terrific assessment tool, also. Previously, it was difficult to know how well the students were doing when only a handful of them were asking questions from a truck-load of exercises. However, when the whole class was engaged on the same few problems, it was easy to walk the room and evaluate their performance and understanding.

The introductory questions naturally lead to the main problem or small set of problems that would drive the lesson. The students were engaged in an investigation, project or activity relating to the concept. Each day my students came to class to solve problems, rather than take notes — a huge change from all the previous “textbook years.” This process of problem-solving and investigation consumed the full class period. Gone were the days of having the students start homework in class. I taught the entire class period.

The homework assignments were only one to three problems long and were typically extensions of the day’s topic, not just practice exercises. I had learned from the international comparisons that America is one of the few countries that pushes the drill-n-kill regime and yet we are at the bottom of the performance pile. So I tried to limit both the number and size of my assignments, and to make them more challenging and contextual.

By doing that, I firmly settled the argument regarding the quantity and frequency of homework that students need to be successful. For the skeptics that are still reluctant to abandon their practice of assigning 30 homework problems a night, I have some strong evidence. My class averages led the district on the district final. With this in mind, I can at least make a case that this new homework philosophy is not hurting my students in anyway.

Another significant change was my lesson planning. Rather than writing examples of how to complete an algorithm or creating cute acronyms to remember esoteric rules, I actually wrote lesson plans. I started planning each lesson by asking: “What do I want the students to know? What is their common misconception of the topic? How can I best get them to understand the topic? How can I challenge them within the context of the topic?” I would then try to create a story/context/scenario and a small set of problems that would best develop understanding of that topic. It was so much fun. This change in my approach to lesson planning was actually a reflection of my new attitude towards teaching. My job description truly shifted from covering material to uncovering knowledge.

Focused, standards-based curriculum; in-depth, problem-solving instruction; short, conceptually-based homework assignments. This experience was so exhilarating that I am now a junkie all over again. I traded my old addiction to the textbook, for a new one — creative lesson planning. This is one habit, though, that I never intend to kick.

All Subjects, Blog

Graph of the Week (New Site)

September 1, 2013 Chris Shore Leave a comment

I am getting the word out on this awesome site: Turner’s Graph of the Week. My friend Kelly Turner did a presentation at the Great San Diego Math Conference last spring and I loved her idea of having students analyze graphs from magazines and newspapers. These are mostly one-quadrant graphs with a natural context. This ties in directly to the Common Core’s call for applications and for reading non-fictional text. I was so impressed that I encouraged her to go public with the idea. I am serving as her megaphone.

Kelly does this activity once a week with her students, thus the name. The site offers several features:

Graphs. You don’t have to find your own. Kelly has already posted 12, and will post more on the GOWS page of the site as the school year progresses.
Submissions. If you like the activity and have graphs of your own that would serve others, email them to turner_k@auhsd.us. Kelly will screen the submissions and build the online collection.
Templates. There is a generic worksheet template with writing prompts to guide students in reading, interpreting and analyzing the graphs.
Samples. On the home page Kelly will offer the graph that she is currently using for the week. Directly below that will be a student sample of the previous week’s graph.

Try it. If you like it, share with your colleagues. The same graph can be used at multiple course levels, with the level of questioning being adjusted to the level of students.

Kelly thanks for all the work on this. You have made teachers’ work easier and students’ education better.

All Subjects, Blog

Tiger Woods Gets a C- in Golf

May 9, 2013 Chris Shore 6 Comments

My district is seriously looking into standards-based grading. I have dabbled in it and see both the value and the pitfalls. Interestingly, I wrote the article below in 2002, long before SBG came into vogue and before the Common Core started flirting with Performance Tasks. While Tiger may not be the top golfer in the world anymore, it speaks directly to my hopes and concerns. I invite some push back here from the SBG gurus.

***********************************************************
Earl Woods? Hello sir, thank you for coming to my classroom to speak with me about your son Tiger. Yes sir, I know that he appears to be doing well at home, but Mr. Woods, to be honest with you, Tiger is in danger of failing golf.

Currently his grade is a C-. I can show you the grade breakdown if you like. Certainly. As you know, there are approximately two hundred professional golfers. Each is ranked in various skill categories. Your son, Tiger, ranks as follows.

Driving Distance	2nd
Driving Accuracy	72nd
Greens in Regulation	1st
Putting Avg.	159th
Eagles	132nd
Birdies	2nd
Scoring Avg.	1st
Sand Save Avg.	4th

As you can see, Tiger does very well in most skill categories, but appears to perform poorly in two. Now, failing in two out of the eight leaves him with a score of 75%. There is a third category in which he is only slightly above average; therefore, he only gets partial credit. This diminishes his seventy-five percent to a 70%, and thus, he gets a C-.

My concern is that if Tiger were to falter in any one of these eight categories, he would surely fail golf. However, there is plenty of room for him to improve in these problem areas. He has an excellent work ethic, so I am confident that with a little more effort, Tiger will succeed. Mr. Woods, thank you for your support in this matter.

Can you imagine ever having this conversation regarding Tiger Wood’s ability as a golfer? How does the best golfer in the world get a near failing grade in golf? The answer is in the assessment.

The rankings given in the previous scenario are true. Furthermore, from this list, the All-Around Rankings of each professional golfer is determined by adding the golfer’s relative rank in each category. The lower the score, the better. Adding Tiger’s categorical rankings places him 10th in the “All-Around Rankings.”

In other words, there supposedly are nine other golfers in the world better skilled than Tiger Woods. Being in the top five percent of all golfers in the overall skill category would certainly raise his grade in golf to at least a B, if not an A. However, he still does not rank as the top All-Around player in the world.

If we change the assessment, though, Tiger fares much better. For instance, Tiger is the richest golfer in the world. He is number on in season earnings and is the all-time career money winner. His is also number one in the World Rankings. The World Rankings are based on how well a golfer finishes in tournament play in comparison with the strength of the field. In other words, how well does the golfer compete?

Tiger wins the most tournaments and wins the most money. In my mind, and that of many others, that makes Tiger the bets golfer n the world. Yet, I am basing my opinion on his performance as a golfer rather than his skill as a golfer. Analyzing two other golfers can show the difference between the value of skill and that of performance. Do the names Cameron Beckman or John Huston ring a bell to you? No? Me, neither, and I am an avid golf fan. The reason that you do not know these names is that these two people are average golfers in the World Rankings. (They don’t win much.) Yet, they both outrank Tiger in the All-Around (2nd and 9th respectively). According to certain forms of assessment, Beckman and Huston are better than Tiger Woods.

Beckman Q2

We can see this scenario being played out in our classrooms. The Beckmans and Hustons get higher grades than the Tiger Woods, because too much of our assessment is based on individual skill rather than on mathematical ability. The Tigers excel in the performance assessments that we occasionally offer, but these are so out weighed by itemized tests that the All-Around Ranking (skill) wins out over the World Ranking (performance),

A more appropriate balance of skills, testing and performance assessment in our classes may send our most underachieving mathematicians to the head of the class.

All Subjects, Blog, Common Core

Rich and Robust

April 8, 2013 Chris Shore Leave a comment

I recently had the pleasure of learning from Tim Kanold of Stevenson High School fame. I heard him speak on several occasions last fall, and he kept saying that we need to involve students in “rich and robust tasks.” He was addressing the Common Core‘s call to the Standards of Practice. These practices can be summarized by saying that the Common Core is demanding students to think and to communicate their thinking. This can’t get done by taking notes from an overhead and doing the odd problems in the textbook. It gets done by purposefully deciding that students are going to solve rich and robust problems rather than simply watch their teacher complete examples of algorithms.

There is still a time and place for direct instruction and guided practice; but that should not be the complete experience for students, which is what we unfortunately find in the vast majority of American classroom instruction. For quite sometime, MPJ has been producing what we hope to be rich and robust tasks. Due to the growth of the internet, the availability of such rich and robust tasks has expanded tremendously. There are many exotic islands of innovation among the seas of tradition, but the blogosphere has made these islands less remote. Below I have listed a few, alongside my paraphrasing of the some of the Common Core Practices. This is not a comprehensive list by any means, however, I encourage you to take a few minutes and peruse these lessons in order to get a quick taste of what I think Kanold means by Rich and Robust.

Persist in Solving Problems. The Coffee Traveler, Dan Meyer
Using Tools. Shop Keeper’s Jar, The Math Projects Journal
Recognizing Patterns. Visual Patterns, Fawn Nguyen
Reasoning Abstractly. Graphing Stories, Buzz Math and Dan Meyer
Mathematical Modeling. Mullet Ratio, Matt Vaudrey

Listed here are some additional sites that offer rich and robust tasks. {Note: I will be happy to update this list with any reader-submitted links, subject to review.}

The activities listed above obviously are not your typical math lessons. For good or for bad, the mathematical frontier created by the experiences highlighted here would make for a far different academic education than the gauntlet of lectures that most of us remember from school.

Now, I am going to assume that while the thought of introducing these large-scale examples into one’s repertoire is exhilarating for many, it may be terrifying for some. Let me ease those hearts by saying that rich and robust can be done on a much smaller scale. For example, we could simply ask the students: “Is x times x equal to two x or x squared. In other words, which of the following statements do you think is always true, if either: x·x = 2x or x·x = x²?”

The CC Practices call for students to construct viable arguments and critique the reasoning of others. If your students stare back at you in silence with this question, then you will know why the Common Core Practices are so needed. If you answer the question for them, then they will watch you participate in a rich and robust activity, while they again participate in mundane note taking. For those that believe that this prompt is too elementary for any course above Algebra, let me assure you that it is not. I posed this very question to my International Baccalaureate students. A handful chose incorrectly, while several “could not remember.” When I asked the rest of the class, which was comprised of some of the brightest seniors on campus, no one could justify their correct answer. The best I got was that they “remember someone teaching us that once.” A simple question turned out to be far more rich and robust than it should have been, but it was a worthwhile day. {Try this one and get back to me.}

I must say here that I am grateful for my math education; it was far better than not having one at all. However, admittedly it was not rich and robust. The question is: Will we make it so for our students? It will take a conscious decision on our part to give our students a different educational experience than most of us had. So ask yourselves: When was the last time that you immersed your students in a rich and robust task? When is the next one planned? Has the time between those two dates been far too long? Are we up to the rich and robust task of offering rich and robust tasks?

All Subjects, Blog

The Elusive Relationship of x & y

October 19, 2012 Chris Shore Leave a comment

“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.

All Subjects, Blog, Boot Camp/Number Sense

Number Tricks (student sample)

October 5, 2012 Chris Shore Leave a comment

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

September 23, 2012 Chris Shore Leave a comment

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.

All Subjects, Blog

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

August 16, 2012 Chris Shore 3 Comments

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.

Textbook Free: Kicking the Habit

Originally printed in The Math Projects Journal in May 2001:

Innovative math lessons you can use in your classroom today