# Math 753 • Session 2

If you’re just joining us, check out the background to this whole experiment, as well as the Session 1 post.

# Summary

We’ll begin with some problem solving and estimation to warm up our brains (complements of MATHCOUNTS and Estimation 180). The majority of our activities in this session will focus on the following two course themes:

#2: Ratios and Proportional Relationships)

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

Additionally, we’ll begin digging into the CCSSM Standards for Mathematical Content (what I’ll refer to from now on as the “CCSSM Content Standards”).

# Resources

Slides

If you want ’em, get ’em here: PDF, Keynote

MATHCOUNTS

We got our wheels turning by working through problems from Warm-Up 2. Need the handbook? Get it here.

Estimation 180

I’ve really enjoyed working through every challenge in my fourth period class this year (we’re on Day 16 on the 16th day of school!). We don’t have enough sessions to do the same thing in Math 753, but I want the teachers to have a sense of the types of challenges we’re skipping over. I’ve provided a two-slide preview of the Day 1-10 and Day 11-20 challenges in the hopes that they’ll be drawn back to them later (either on their own or with their students).

Our challenges for Session 2: Day 13 and Day 14

CCSSM Content Standards

Our first real venture into the content standards for CCSSM. After briefly discussing the Four Big Ideas in Algebra conversation that started with Grant Wiggins’ 100th blog post, teachers will work on this:

The Running Game

I’m excited to bring the next pair of Running Game challenges to the class, partially because the challenges increase slightly in difficulty with each pair of days, but also (and primarily) because I have a shiny new handout.

Our scheduled challenges: Day 3 and Day 4

Visual Patterns

In the first session we explored several proportional relationships. (Check the Session 1 slides for specifics.) In this second session we’ll branch out to look at patterns involving a steady rate of increase with a slight shift away from simple multiples. For example, instead of 3, 6, 9, 12, etc., we might look at 4, 7, 10, 13, etc.

If that makes no sense, check out the Session 2 slides.

In last week’s session I introduced a half-baked task based on caloric content of beverages at the In N Out soda fountain. The task was mediocre, but the context (in my opinion) had some merit. With that in mind, I’ve revamped the task. The focus now is on making connections among multiple representations.

The slide deck contains a few potentially useful images, but the real goods are here:

Students will work in small groups (2 to 4, ideally) to cut out and then match the various representations contained in each packet of “ingredients.”

Graphing Stories

Water volume (by Esteban Diaz-Ibarra) and Distance from center of carousel (by Adam Poetzel).

Compliments of course to Dan Meyer and BuzzMath for the excellent resource.

Grant Wiggins (author of Understanding by Design) recently started a conversation, in his 100th blog post, no less, about the big ideas in algebra. The key passage:

Here is a thought experiment: 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? Doubt it. Because algebra courses, as designed, have no big ideas, as taught, just a list of topics. Look at any textbook: each chapter is just a new tool. There is no throughline to the course nor are their priority ideas that recur and go deeper, by design. In fact, no problems ever require work from many chapters simultaneously, just learning and being quizzed on each topic – a telling sign.

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

1. Read the post and all of the comments. (Get a beverage and a snack ready; there are quite a few.)
2. Spend at least 24 hours with the ideas jostling around in your brain.
3. Add your own voice to the conversation by posting a comment, either on Wiggins blog, or here.
4. Things to discuss might include (a) your own list of 4 big ideas, (b) ways in which your list is being reshaped as a result of our class and the discussion started by Grant Wiggins, (c) questions you’ve always had about the big ideas in algebra, (d) questions you never knew you had until now, and (e) anything else that comes to mind as a result of the reading assignment.

Bonus task (this kind of bonus, not the “points” kind):

• Never used Twitter? Get your toes wet by exploring Grant Wiggins’ timeline. Keep your eyes out for new threads and clarifying comments in the “big ideas in algebra” conversation.
• Don’t worry if you get distracted and fall down a few unrelated rabbit holes. Part of the beauty in the conversations on Twitter is that you can find millions of different topics, discussed at varying levels of intensity, and they’re often just a click or two away.

1. Charlene Jovel says:

While reading Grant Wiggins I could not help but smile and agree with so much of what he was saying. I had the experience of going through Algebra courses here in the US and in El Salvador. I had to take algebra twice not because I did not show proficiency the first time, but because the US would not honor my Algebra course from El Salvador which added to my frustration with the subject. I have always questioned why Algebra seemed so abstract and pointless. I could not identify the big general ideas of the course. I think that if you would have asked my 14 yr old self what Algebra taught; I would have responded, “how to isolate X.” Which I now understand is not the point. So why is Algebra such a pointless subject?

I agree will what Wiggins had to say, it is simply taught out of context. Now my questions are many. So really, What are the big ideas in Algebra? What does Arithmetic do, that Algebra can’t and vice-versa. Should Algebra be taught as a focus of patterns and generalizations? After reading the blog I’ve decided that one of Algebra’s big ideas should be Real World Application.

I could not help but wonder the following:

2. How does Elementary (K-6) foster the dislike for mathematics?
3. How can I as a fifth grade teacher help my students be prepared for Algebraic Thinking?

2. Daniel Rocha says:

Not sure if “this: is where we were supposed to blog about Wiggins bashing of Algebra. As a 4th year teacher of MIddle School math, its amazing at how much I don’t know about math. My Master teacher during my credential program told me I would one day be amazed by how much I didn’t know. And boy was he right.
The question that I am left with would be: If you can teach Geometry by building a desk, how would you teach Algebra. Recently we were taught how to solve algebraic expressions using tape diagrams. Is this satisfactory? I don’t believe it would be since it may not make the connection to real life situations. So I am left wondering, am I doing this right?

3. Teri Simas says:

For nine years I have been “one of those” – someone who has taught 8th grade Algebra I in frequently discrete and abstract chunks, pieces and isolated skills. One of the biggest laments from my students over the years has been the question, “when will we ever have to use this?” And I do wish that I would always have had an interesting answer for them to that question, but sadly I have not. I have personally seen how the traditional algebra course can drag the joy and pleasure of learning from a student. My own reaction to the disinterest of my students in the subject has been attempts to give them more opportunities to investigate and learn through inquiry whenever I could, hoping that increasing the discovery of ideas on their own would develop a greater investment and interest in what they were learning. I have struggled a lot to find opportunities in the classroom to show students where they could see where connections could be made for the out of context skills we were covering, but sadly I have not always been successful.
The first time I read the then proposed common core standards for 8th grade I was very excited, because here now there appeared to be movements to lessen students’ disconnect, making the subject more meaningful and the learning more enduring. A teacher who had been teaching at our district’s middle school began teaching 7th grade math and science at my school this year. Last week she said one of the kids on leaving her science class surprised her by saying “I had fun today.” Another teacher asked her “Haven’t you ever heard anyone saying that before?” Her response was simply, “I taught Algebra.”
I am very excited to be teaching 8th grade math right now. This time of transition is energizing – and I KNOW that more of my kids will become turned on to math because of the revisions and shifts in instruction that are going on now. The Standards for Mathematical Practice throughout the levels of education will definitely provide a driving force that will improve and deepen attainment of knowledge, and hopefully also increase students’ affection for mathematics. Change is in the air . . . and it can only be good!

4. Betty Kennemur says:

As I read Grant Wiggins post I was reminded of my husband’s experiences with Algebra – he had passed it, barely, in high school, but then ran into a need for college algebra as a prerequisite for his professional courses. I was the nerd who had always enjoyed solving algebraic equations, so managed to help him understand enough to (barely) pass that course. His constant lament was that nothing made sense, and it had no purpose (sounds vaguely familiar!) Many years later, we were in a situation where he was working as a reserve firefighter. I will never forget the day he came home and informed me that he had finally figured out how to solve an algebraic equation – because there was a reason to do so. The engineer’s equation determined the water pressure available to various hoses, and only if that was solved correctly could they depend on being able to operate their equipment efficiently. Suddenly, with a very practical reason to solve this rather complicated equation, he could do it just fine.
I have to admit that when my 7th grade pre-algebra students complained “When are we every going to use this?” I don’t think I ever replied “Just wait until you’re a fireman!” My standard reply was “You’re going to need this in Algebra next year.” Pretty lame, I know, but – aside from the fact that the question was usually asked at rather inopportune times – I usually didn’t have a better answer for them. That’s part of the reason I took the opportunity to move back to 6th grade. I am excited about the conversations and the move to make everything we do as practical and relevant as possible, and would echo Teri’s comment above: “Change is in the air . . . and it can only be good!”