New here? Check out the background to this series before you dive in.

# Summary

Once again we’ll get things rolling with some MATHCOUNTS problem solving and Estimation 180… er, well… estimation.

For at least one more week the primary focus will be on the following two course themes:

#2: Ratios and Proportional Relationships

#5: Four Representations: Numerical, Graphical, Algebraic, Verbal

We’ll also continue the “Four Big Ideas in Algebra” conversation.

# Resources

**Slides**

**MATHCOUNTS**

After two weeks of Warm-Ups we’re ready for our first Workout. Still need a copy of the handbook? It’s right here, it’s awesome, and it’s free.

**Estimation 180**

Nothing says *mathematical reasoning* like a couple rolls of toilet paper. In Session 3 we’ll work on Day 28 and Day 29 on the site. Here’s a look at the rest of what Days 21-30 have to offer:

**Four Big Ideas in Algebra**

We’ll spend a few minutes discussing our reactions to the Session 2 reading assignment (a Grant Wiggins blog post), including any affirmation, surprises, disagreement, new insights, or new questions that came up in the course of reading the post and its comments, and writing a comment of our own.

For now, we’ll leave the larger “Four Big Ideas” conversation alone until a later session.

**The Running Game**

I’ve tweaked the new handout a bit based on a comment from the class in the Session 2 Feedback Form. Here’s the latest and greatest version.

Our scheduled challenges: Day 5 and Day 6

**Visual Patterns**

No time for a new style of visual patterns (we’ll continue pressing forward in a week or two). For now I want to camp on linear growth a bit longer. Instead of giving a few new challenges from the website (or of my own invention), I have a “create your own” assignment for each member of the class. Details below.

**Automatic Change Dispenser**

On my way through the checkout at a local supermarket, I saw this:

I think the MTBoS is sharpening (or further twisting?) my brain, because I instantly wondered…

What’s the total value of those coins?

So I asked this on Twitter:

I’ll add a full length blog post about how my thoughts grew from “Hey, this is a cool estimation task” to “Whoa, this could be a pretty sweet three act task if I don’t blow the presentation (like I usually do).” For now, check out the resources I’ve posted for this task over at 101qs.

**Desmos Proportion Graphing Challenges**

Earlier this year Dan Anderson, Justin Lanier, and I launched a website called Daily Desmos. Each day we (or someone else from our awesome and growing authoring team) creates one basic (er, well, not so basic) and one advanced graphing challenge. They’re fun to make and fun to solve.

(For more background, go here. For details about what’s next for Daily Desmos, go here and here.)

We recently had several discussions about how to make Desmos more useful to classroom teachers. One idea: Create a series of related challenges, intentional in sequence, progressing from simple to more challenging, and in doing so provide students with a sandbox for developing their graphing skills in an enjoyable, dynamic, exploratory (yet still somewhat structured) environment. (Sorry about that last sentence; it got a bit out of control.)

At any rate, because one of the major themes in Math 753 is proportional reasoning, and because we’ve been discussing creating a sequence of linear graphing challenges, I created a sequence of *proportional* graphing challenges (some might use the term *direct variation*). I’m waiting for feedback on the quality (or lackthereof) of this sequence. Once we’re happy with the quality and format, we’ll turn our attention to a linear sequence, then (probably) quadratic, and so on.

**Graphing Stories**

Since we ran short on time in the first two sessions, I want to revisit the first four stories in Session 3. Soon we’ll continue moving forward, and soon after that we’ll have an assignment where each of us (myself included) has to create our own 15 second graphing story (or two).

# Reading Assignment

We’ll continue with our “Big Ideas in Algebra” discussion by reading a few responses to Grant Wiggins’ original post. Read the following (including all comments):

- Four Big Ideas in Algebra (by Patrick Honner, @mrhonner)
- Pretty Big Ideas (by Chris Lusto, aka @lustomatical)
- Pretty Big Ideas for Intermediate (Highschoolish) Mathematics (by Kate Nowak, aka @k8nowak)
- Grant Wiggins’ response to Patrick’s post

**Your task (if you’re in Math 753, or following along at home):**

- Read the four posts above, including all of the comments.
- Spend at least 24 hours with the ideas bouncing around your brain.
- Add your own voice to the conversation by posting a thoughtful comment, either on one of the blogs above, or on this post.
- Things to discuss might include (a) your latest list of four big ideas in algebra, (b) ways in which your list is being reshaped as a result of reading several more perspectives, (c) new questions or confusion you have about the big ideas in algebra, (d) new clarity you have about anything related to this discussion, and (e) anything else that comes to mind as a result of the reading assignment.

# Visual Patterns Assignment #1

Create one or two of your own visual patterns. They may be inspired by what we’ve done in class, or what you see over at Visual Patterns, but they must be your own invention. A few more details:

- Create a visual for Step 1, Step 2, and Step 3. (you may include Step 4 if you find it helpful or necessary.)
- Create each step by carefully drawing, using a computer, or taking photographs of patterns you see or build in the physical world. (I would love to see the latter.)
- Put Steps 1-3 (or 1-4) on a single sheet of paper (physical, digital, or both). Bring
**at least four physical copies**of your visual pattern to class next week. - Ideally, your pattern will fit the linear growth theme we’ve explored in the first few sessions.

“Can you identify 4 big ideas in algebra, ideas that not only provide a powerful set of intellectual priorities for the course but that have rich connections to other fields?”

There were a number of questions that came into my mind when Grant Wiggins proposed this challenge:

Does he already have the answer in mind?

Why 4? Why not 5 big ideas, or even just 1?

Other fields?…like physics? Business? Engineering?

Is he really even interested in the answer, or is he just proving that no one ever bothers to contemplate the real reason why algebra is uniquely important?

It’s made me think. I’ve had to look up the basic definition of algebra. It’s surprisingly brief. Most sources state that it’s a system of equations that uses variables to solve problems of unknown quantities. That’s it. So what is Grant Wiggins trying to do?

It’s been interesting to see how some of the discussions about Wiggins’ task have taken a different tack. Soccer? Creating and collapsing space in the soccer field is a big idea? That sounds more like a strategy. The big idea in soccer (and in most other games) is to make goals, to win the game.

The one statement that resonated with me about algebra was a quote in a post by Max Ray:

“…math is about “Creat[ing] structure when you’re building; look[ing] for structure when you’re exploring.”

Add to that a response to a post from Grant Wiggins:

“I always ask algebra teachers this question: do your students know what algebra does that arithmetic can’t do? Do your students know what analyses algebra enables that can’t be done with basic arithmetic?”

Therefore, my quest would be this; what will algebra do, that no other discipline can do, that will enable me to make sense of God’s creation?

I have never played soccer, but I have been the typical soccer mom . . . driving my kids three days a week to practices, cutting up the oranges for the snacks, carrying a beach umbrella for shade when it was hot, and huddling around a portable heater when it was cold. I have watched A LOT of soccer over the years. Here are some other analogies between the game and algebra that I think can also be made:

1. You need to know the rules of the game to play. For example, keep both feet on the ground when you throw in the ball.

In some way the processes and conventions of algebra need a role as a big idea. They are the structure that makes the solutions possible. As a teacher (or in a continuance of the soccer analogy a “coach” of algebra team players) I want to make sure that the instruction of these “rules” is in a way that the students will want to participate and become eager to play the game. To make this possible I want to give them opportunities to see real-world applications of these skills. I think the second part of the idea “creating structure when you are building, looking for structure when you are exploring” is something to consider during instruction because if things are presented to kids as a challenge to investigate they will be much more caught up in what they are learning. I want to give them chances to explore and learn through their own inquiry as much as possible.

2. Two teams play on the field. Know the boundaries of the field. To keep the field balanced your whole team needs to move as one.

I have always thought of algebra as a sort of balancing dance. One of my favorite phrases to use with kids in class is “When you do it to one side, you have to do it to the other.” I am not sure it this is a big idea on its own or if it really is a part of the first one – a compartment of the processes and rules of algebra – but to me over the years though it has seemed to have been in large part the essence of what we do in finding solutions algebraically.

3. When something happens on the field with the opposing team, react. Don’t just stand in one place.

At the heart of algebra we are looking at the relationships between and amongst quantities. As one set of values change, the others do too. Look for patterns. Understand the relationships between values. Know that with equations things are not static.

I am still stumped for a fourth big idea. If I look at this as a directive for how to reach kids and make them eager learners of algebra then making connections to what is being taught is critical. Maybe the idea that algebra models and represents actions of the world numerically is a part of what I am looking for here. I am not sure it is a whole lot different though than my third big idea.

I never thought that Algebra could spark so much emotion. As I read the different posts I’m still confused as to really what a big idea is. At first, I thought it was an overarching theme that keeps coming up in the subject but then WIggins took that away. I believe some of my challenges with coming up with four main ideas is my own insecurities about Algebra.

I do not teach seventh or eight grade therefore some of the vocabulary thrown by Patrick was completely confusing and he got me lost. Then Wiggins brought it back when he said that big ideas should be understood by novices and experts alike. I decided to look at the big ideas of Algebra by just looking at the standards that I have to teach my fifth graders about Algebraic thinking. My fifth graders are required to understand the distributive property, Order of Operations, simple equations, and graphs. When I think of big ideas in Algebra I always come back to the fact that all this crazy equations have meaning. The equations are representing a story. WIth that said a big idea of algebra for me would be that values can be represented in different ways. My fifth graders learn that a variable takes the place of the value we are trying to solve for. I’m also thinking that patterns is one big idea in Algebra. If it wasn’t for mathematical patterns and observation, we wouldn’t come up with formulas to solve problems. Formulas are equations that were written based on a prediction about a certain pattern. A graph illustrates a pattern; therefore, an equation does too.

I’m still trying to wrap my head around the idea of Big Ideas but so far I think I have two. So are big ideas just main concepts that everybody should understand? Are we making these big ideas to difficult by over analyzing it?

As I read and re-read the blogs and their responses, I feel less and less qualified to jump into this discussion at all! There are, however, a few things that stood out – and even made sense to me:

1) Within the soccer context, Grant commented that “The game is problem solving.” I don’t remember much of my high school algebra (perhaps because there really isn’t a lot that the average person uses regularly????) but I do remember that what I always enjoyed about algebra was just that – solving the equations seemed like some kind of big puzzle or game. Everyone enjoys a good game.

2) Several comments mentioned context and relationships – it seems that relationships in math are much like relationships in life: we really can’t do much in isolation, and neither can our numbers and variables.

3) Coming from the create structure/look for structure concept, Max Ray’s list of important things for math students to do included “Be as concise and organized and precise in your notations as is humanly possible.” Being organized and precise (i.e.: structured) seems to also be an important life skill, both in maintaining our important relationships and in accomplishing tasks that need to be done.

So . . . maybe, even if the content of algebra is not all that practical (going back to Grant’s original bashing of the subject and its requirement for high school graduation), the very process and discipline involved in learning those skills and tools are actually beneficial in preparing students for “real life.”

It seems to me that the original point that Grant was trying to make is being obscured by the argument. Grant’s point was that Algebra as a class is too convoluted and abstract to be easily mastered by the general populous and thus became a guardian to ‘keep out the riffraff’.

Now people are arguing over what constitutes a Big Idea. What idea is too big to be a Big Idea? That doesn’t even make sense. There are themes throughout mathematics that can be traced through grade levels. Number sense for example. Since it takes more than one year to cover all the aspects of number sense it is too big an idea? No, clearly not. Number sense must be learned as a person matures and begins to grasp concepts that they weren’t capable of before.

I think that the point that was made at the beginning is being lost in this discussion. The way that we teach Algebra (and for that matter most mathematics beginning in about 6th grade) is broken. We have lost our focus on teaching mathematical literacy by example and we have turned to teaching process. Mathematics has been boiled down to a set of rules to be memorized and rehearsed rather than explored.

Algebra is a tool, much like a wrench is a tool. If you asked a mechanic to take a course on the study of wrenches in which you analyzed the properties that make up the wrench and how it is formed you would loose many capable mechanics. If you teach a course where you use the wrenches to fix cars, and in doing so become familiar with the form and function of the wrench (as well as why you use a particular wrench for a particular task and why that wrench is made from the material it is made from and so on…) you produce a mechanic who has the ability to use his tools affectively. That is what we should be attempting to do with Algebra.

If we were to search long enough I do think that there is enough material in Algebra to come up with several Big Ideas that are worthy of learning in and of themselves. We certainly can search for Pretty Big Ideas and find them in Algebra as Chris Lusto would suggest. I am inclined to agree with Horner when he says that, “formal abstraction is an important theme of the course.” The ability to take a real world situation and express it as an equation should be a goal of any Algebra class. No matter what ideas you come up with the merits of them can certainly be debated, and it is open to personal opinion if they meet all the requirements.