Tag Archives: problem solving

Introductions & Neuron Facts in Algebra 2

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Day 2, Thurs Aug 11, 2016

The Brain Surgeon
Today, we began my regular routine of designating a daily Brain Surgeon. Since this was our first day of the Brain Surgeon, I introduced the routines of the Drum Roll, Reading of the Dual Target, Music Cues, and the Wrinkle Sprinkle. The students seem to embrace the spirit of of it all.

Student Introductions
As with every new school year, I had each student briefly state their name and something interesting about themselves. When they were all done,  I recited all their names. That always impresses a class. Then I told them things about myself. I state that yesterday we started with math, because that is what we are all about here. But since I teach math to them, they are also important and I need to know who they are.

Growth Mindset
Most of our Course Teams across the district agreed to do some kind of growth mindset activity. Here was mine.

I started by summarizing the plethora of lists of fixed vs growth mind set statements with two pictures. I told the students that research in student learning is showing that self-perception of talent as a limit or as a starting point has a tremendous influence on their learning.

Talent Wall

Then I shared that scans of the brain of someone with a fixed mindset versus a growth mindset, shows something very interesting. When faced with a challenge, the fixed mindset brain “goes cold.” It literally shuts down. However, when faced with the same challenge, the growth mindset brain “fires up.” It knows that more is being asked of it, so it kicks into high gear to meet the challenge, rather than duck it.
Brain MindsetsNow it was time to test out where we see ourselves demonstrating  a fixed or growth mindset.

Neuron Facts
I gave the students the worksheet with the Neuron Facts on the front side. I found these on the internet and thought they would make for a good lesson since they highlight the amazing function of our brains. I added the subheadings of Fast, Crowded ,etc. I started with a common practice of mine Notice & Wonder popularized by Annie Fetter (@MFAnnie) of Math Forum.  My Gradual Reel-In process looked something like this:

  1. You Do: Independent response.
  2. Ya’ll Do: Each member of the group shares both their notice and wonder.
  3. We Do: Each group decides on one Notice and one Wonder from those shared. These get shared out by each group as I write them on the board.
  4. I Do: I summarize the major point(s) that I want all students walking out with. Here it was the process of Noticing and Wondering and how we facilitate group discussion in class… And of course how amazing our brains are.

The groups were then tasked with doing one problem together. Homework was to do one more.

Wrinkle Sprinkle
Tying into the concept of the plasticity of the brain, I joke that when we learn we get a new wrinkle on the brain. Each class then concludes with what we learned that day. The brain surgeon leads and records the discussion. The students today stated that they learned…

  • Negative thoughts shut down your brain
  • Speed of the brain cell
  • The amount of oxygen the brain uses

Get to the Core of The Core

apple coreThe Common Core curriculum can basically be summed up in the following sentence:

Teach your students to THINK and COMMUNICATE their thinking.

Thinking and communicating are the 21st Century skills. Many people believe that the skills of the future involve the competent use of technology. While it is true that using digital tools in school and the work place is the new reality, it is actually the proliferation of technology that makes thinking and communicating imperative in the information age. When all the knowledge of humankind is available at anyone’s fingertips, memorizing information becomes far less important than being able to construct, evaluate and apply it. You can Google information; you cannot Google thinking.

So the core of the Core truly is Thinking & Communicating.

To make my case for this, I would like to pose that the following equation

6 + 4 + 4 + 8 = 22

be adjusted to

6 + 4 + 4 + 8 = 21

Before you start shouting that everything you have read on Facebook about the Common Core is true, let me declare that I am using this equation simply as a teaching device, not a true mathematical statement. You will understand what I mean after I present my evidence.

6 Shifts

Let me start my case that the core of the Core is Thinking & Communicating with the 6 Shifts, which are best represented by the following document found at Engage NY.

6 Shifts

In essence, these shifts are redefining rigor. Old school rigor was defined as sitting quietly taking notes, and completing long homework assignments in isolation. The new school definition of rigor envelops the last 4 shifts on the list: Fluency, Deep Understanding, Applications, and Dual Intensity. The rigor is now placed on the students’ minds instead of on their behinds.

The shifts are also calling for balance. Dual Intensity insists on both procedural fluency AND critical thinking by the students at a high level. It is not about dual mediocrity or about throwing the old out for the new, but a rich coupling of both mechanics and problem solving.

Therefore, I make the case that:
                     6 Shifts = 21st Century Skills,
which are to
                     Think & Communicate.

4 C’s

Another list that is framing much of the Common Core dialoge is the 4 C’s. Resources for this list can be found at Partnership for 21st Century Learning (p21.org).

4 C'sThese C’s redefine school…

The old school definition: A place where young people go to watch old people work.

The new school definition: A place where old people go to teach young people to think.

… and they redefine learning.

The difference of old school vs new school learning can best be contrasted by the following images of the brain.

Brain Chillin   Brain Build

The image on the left shows a passive brain that just hangs out as we stuff it with esoteric trivia. The image on the right shows a brain being built, symbolizing its plasticity. We now know that when the brain learns, its neurons make new connection with each other. In other words, learning literally builds the brain. The 4 C’s  claim that this building involves the capacity of the students’ brains to Critically Think, Communicate, Create and Collaborate.

Therefore, I make the case that:
                     4 C’s = 21st Century Skills
which are to
                     Think & Communicate.

4 Claims

Smarter Balance creates it’s assessments based on 4 Claims. (I teach in California. PARCC has 4 Claims that closely align to those of SBAC.)


4 Claims

Notice that Claims #2 & 3 are explicitly stated as Thinking & Communicating, which also overlaps with two of the 4 C’s. Mathematical modeling is #4, which will be discussed later. I want to point out here that Claim #1 reinforces our idea of Dual Intensity from the 6 shifts.

There are two important notes for teachers about this first claim. 1) It says Concepts and Procedures, not just procedures. The students need to know the why not just the how. 2) The Procedures alone account for about 30% of the new state tests, so if we continue to teach as has been traditionally done in America, we will fail to prepare our students for the other 70% of the exam which will assess their conceptual understanding as well as their abilities in problem solving, communicating and modeling.

Therefore, I make the case that:
                     4 Claims = 21st Century Skills
which are to
                     Think & Communicate.

8 Practices

If you open the Common Core Standards for Mathematics, the first two pages of the beastly document contain a detailed description of the Standards of Mathematical Practice. Then at the beginning of each of the grade level sections for the Standards of Content you will find 8 Practices summarized in the grey box shown below.
8 practicesWhat do you notice about the list? Indeed, these habits of mind all involve Thinking & Communicating. While the content standards change with each new grade level, the practice standards do not. With each year of school the students are expected to get better at these 8 Practices. Notice that the first half of the list has already been included in the ones discussed previously: Problem Solving, Communicating Reasoning, Constructing Viable Arguments and Modeling. A case is often made that the other four are embedded in these first four. However one might interpret the list, “Memorize and Regurgitate” is not on there.

Therefore, I make the case that:
                     8 Practices = 21st Century Skills
which are to
                     Think & Communicate.

The Sum of the Numbers

So, as you can now see, the 6 Shifts, the 4 C’s, the 4 Claims and the 8 Practices are all focused on the 21st Century Skills of Thinking & Communicating. Therefore, I can finally, explain my new equation …


    6 Shifts
    4 C’s
    4 Claims
+  8 Practices
= 21st Century Skills

then 6 + 4 + 4 + 8 = 21!

None of these numbers represents a list of content, because the content changes brought on by the Common Core, while significant, are actually no big deal in the long run. A few years from now we won’t remember all the fuss regarding Statistics and Transformations, but we will all spend the rest of our careers learning how to teach kids to Think & Communicate.

I rest my case.

Dr. Jon Star Speaks HOT Heresy

Pic StarDr. 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.

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…

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.

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.

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.

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.



Multiple or Best Reps?

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.

Ultimate Cosmic Power in an Itty-Bitty Thinking Space

“Give me any combination of two numbers that have a sum of seven,” I said to my students. One person offered, “Two, five,” which I wrote on the board as (2, 5). I asked for a few more and got (5, 2), (1, 6) and (0, 7).

“Good,” I praised, “now give me ALL the combinations of two numbers that have a sum of seven.” They chuckled. “I want them all, and I want you to write them down.” The students were hesitant, because they knew there are an infinite number of pairs that have a sum of seven. So I challenged one of them to a race. “You write them down on your paper, I’ll write them on the board. Nobody goes to lunch, until one of us is done. Ready, Go!” I scribbled on the board x + y = 7. “Done!”

They didn’t buy it. “I have just written ALL the combinations of two numbers that have a sum of seven. Since you don’t believe me, I’ll do it a different way. In fact, I’ll take you all on. All of you write down combinations of numbers, that way you get done in one-thirtieth of the time, and I’ll still woop ya. Ready, Go!” I quickly sketched the graph of x + y = 7. “Done!”

This goofy little exercise was intended to impart the idea that mathematics gives us the ability to represent an infinite number of elements in a finite time with little effort. I spread my arms wide in front of the class and exclaimed “Ultimate cosmic power…” then brought my hands to rest on a student’s head and continued, “…in an itty-bitty thinking space.” (A play on the Genie’s words from the Disney movie Aladdin.) No offense to the student, but our brains are not very big. Yet, we were able to take all the pairs of numbers whose sum is seven, shove them all in our heads at once and think about them all at the same time. The ability to then communicate them to the world outside of our heads using equations and graphs is what makes mathematics the Ultimate Cosmic Power.

However, most people don’t share in our awe of this power. I believe that is because we never initiate them into our mathematical club. We keep students on the other side of the room while they watch us speak club code and give the secret club handshake, but we never let them in. I have proof of this …

Thoughts on Math by the Uninitiated

From an Algebra 2 student who was just kidding, but his joke reflects how many people perceive the purpose of math:

 Algebra would be a lot easier if they just told you what x was.
– Scott, Class of ’94

For a moment, I thought this next one was kind of cute when a student had just simplified 3x + 2x to equal 5x:

Only in math do you put two things together and get a smaller thing.
– Neal, Class of ’99

Then I realized … In math we don’t combine to make a smaller thing; we combine to make an equal thing!

Then there was the English Teacher who stopped me in the hallway one day, visibly irritated, and poking me in the chest:

You math teachers aren’t very good. My whole life you have been asking people to find x. Why can’t you find it yourselves?

I don’t know what set her off on that day, but I do know one thing about the three people who made the statements above. They all see the sole purpose of math as a cognitive Easter egg hunt. They close their eyes, while the teacher hides the variable. Ready, set, go. Praises and smiles when the basket is full.

They have no appreciation for the true beauty and power of Algebra, because we never initiated them into the club. So how do we teach them the secret handshake?

The Initiation Rites

  • Context
  • Multiple Representations
  • Complexity
  • Abstract Generalizations


Too often we jump straight to the naked math problem. For example,  we ask students to graph y = 2x + 1 or evaluate 2 – 5, without offering any kind of context. Context gives the students something to cognitively hold onto while they are grappling with the math concepts. Take for instance, the teaching of negative integers. In the Wallflowers lesson, students are asked to mathematically represent scenarios that they can relate to (a high school dance). While these scenarios are a bit contrived (girls are positive and boys are negative), students can “see” how a balance of positives and negatives equals zero, taking away negatives leaves behind positives, etc. From here it is useful to go to other contexts that are more applicable. The Postman Always Ring Twice relates positive values to checks and negative numbers to bills, showing how truly “subtracting a negative is the same as adding a positive.” By presenting the context first and then asking the students to represent it symbolically, we give them a framework in which to think when eventually presented with the naked math problem.

Multiple Representations

Thoughts on Math by One of the Initiated

I once met a Calculus teacher in Massachusetts who was originally from India. She was very distraught about teaching in America, She said that she kept getting complaints from both students and parents about her teaching style. She said that all she was doing was teaching the way they do in India. When I asked her to characterize the style for me, she said that in India there is a saying:

If you know how to do one problem inside and out, you can do a hundred just like it.
– Seheti, Math Teacher from India

I could easily see the contrast. In India they teach students a hundred ways to do one problem. In America we teach one way to do a hundred problems.

To further make Seheti’s point, I had a conversation with my daughter’s third grade teacher on this very point. She asked me:

Thoughts on Math by the Uninitiated

How would you do this problem?

Of course, she gave me this example because it requires multiple borrowing in a traditional algorithm. I told her, however, that I don’t see it as a subtraction problem; I see it as an addition problem. I have 165 above 3,000 and 13 below 3, 000, therefore, 165 plus 10 plus 3 was 178 … with no borrowing. Her response was, “But why would you go through all that trouble?” I chuckled at the unintentional irony, placed the pencil on the paper and challenged, ” Do your method without picking up the pencil.”

These examples show the strength in teaching students Multiple Representations of a problem, and the weakness of teaching only one method. In the Candy Bars lesson, multi-link cubes are used to demonstrate why we need a common denominator when adding fractions. Fractions, after all, are merely relations to the whole. Operations on fractions can only be done then on the same size whole!

When students are first asked to show one-half of a candy bar with the cubes they put together two cubes of different colors. When asked to “build” two-thirds of a candy bar with the cubes, the construct a stick of three cubes, one being of a different color. When prompted to “show” what fraction of the candy bar they have if they are given portions, they naturally connect the two to make one, which shows two-fifths, which is exactly how they incorrectly complete the algorithm for adding fractions: they add both numerators and denominators.

When corrected and told that the bars must be the same size and still show one-half and two-thirds, the students independently build sticks of 6 cubes each (three of one are colored, two of the other are colored). When then pressed to now tell us how much “of the same size candy bar they now have, they combine the colored cubes, but keep the stick the same 6-cube length and present … five-sixths, which is correct. Publically generating an algorithm that now represents what we do is easy, and the cubes offer a model for students to fall back on in the event that they forget the procedure.


Too often we think that our job is to always make math simple for our students. This initiative then leads us to break problems down to their tiny, separate parts, and we never ask the students to put it all together. This point was made by Tad Watanabe in NCTM’s Mathematics Teacher (Vol 93, No 1, p 31). Mr. Watanabe showed a high school entrance exam from Japan. It had only 7 problems. One of them is displayed on the right. (Click to enlarge the image.)

Take some time to do the above problem. It puts to shame what we expect from our 8th graders: “There are 10 marbles in the box. 3 are red. What is the probability of drawing a red marble?” The Japanese expect there students to do complex problems, therefore, they teach them to do complex problems. We Americans often feel that we have failed if we pose students with difficult problems. One would have to look long and far to find an Algebra final exam in the United States with the level of complexity of the Japanese example above. It is not that Japanese students are more capable than American students and therefore, they can do these kinds of problems. The Japanese students are more capable because they are regularly asked to do these kind of problems!

To further the point on Complexity, I would like to share the story of the American math teacher who visited schools in Bulgaria. When asked to contrast math education between Europe and the U.S., he said he could do so by showing a typical question from both countries:

Typical Geometry Question in Bulgaria: Draw a triangle. Draw a semicircle on each side. Within each semicircle , inscribe the rectangle of the greatest area. Draw the lines that pass through the centers perpendicular to the side of the triangle. Prove that these lines are concurrent.

Typical Geometry Question in America: Draw a triangle.

Abstract Generalizations

Finally, the ultimate in Ultimate Cosmic Power: Abstract Generalizations. In other words, students are asked to model their world mathematically. As when I ask my daughter and her friend when they were in third grade, “Your class has 20 students. How many are boys and how many are girls ? How many boys and girls might be in another class of 20 students?” We went through several scenarios, and then I asked, “If we allow b to represent the number of boys, and g to represent the number of girls in a class of 20 students, what would you say about the the number of boys, girls and the total students?” They said “b plus a equals 20.” I then showed how to write b + g = 20, and they agreed. There were several abstract representations going on with the girls. Words and an equation, and a brief encounter with data.

The Rising Water lesson does the same thing, but more formally. It first poses a context in words (one representation): A swimming pool contains water 10 cm deep. The water is rising 3 cm per minute. The students are to then generate a table of values, an equation and a graph for this scenario (3 more representations). The objective of the lesson is to teach students that all four representations describe the same relationship between two quantities (time and water depth). The students are then asked to generate their own scenarios, with the four representations. The more that students are asked to create their own models, the better capable they will be when they are presented with one.

So our Initiation Rites into our math club are these four components of Ultimate Cosmic Power (Context, Multiple Representations, Complexity and Abstract Generalizations). We will go back to these constantly in our discussions, as well as to the Four and a Half Principles of Quality Math Instruction posted previously. To show that there is hope in teaching in this manner, I share with you a statement from a former student at the end of the year.

Thoughts on Math by One of the Initiated

Poetry is the language of love. Math is the language of everything else.
– John, Class of ’99