# Project/Journal 11

This was the year’s largest and most technical writing assignment. I required students to submit drafts which I edited and returned to them so they would have the chance to make their writing more precise. The project consumed a large chunk of our time, but the process of explaining their methods and results in writing, getting a draft edited, then trying to make their language more precise was a turning point for my students’ mathematical communication.

All but one of my students were seniors and were pretty well invested in learning about student loans. On Day One I introduced subsidized vs. unsubsidized student loans. (The project presents case studies using unsubsidized student loans.) I also addressed how student loan interest is actually calculated* and explained that A=Pe^(rt) would give good enough approximations. In the mish-mash curriculum I worked from, A=Pe^(rt) is used as an application for the natural log, so I wanted to give students experience using that formula even though the unit was not connected to other types of interest growth.

The project gives 3 cases:

1. You take out an unsubsidized student loan of \$3000 your first semester of college and it grows while you attend school for 5 years.
2. You take out that same loan but you make an early payment of \$500 two years into college.
3. You take out the same loan but a semester later.

*09/10/14: An improvement might be to ask students to give estimates or judgments on the three options before any calculation.

The first day’s assignment was to calculate what the debt would grow to by graduation in each case and estimate the cost of repayment. I left students pretty much to their own devices to figure it out, which frustrated them to no end.

On Day Two we compared their solutions, went over the correct solutions, and they worked on the first two paragraphs of the write-up, turning in drafts the next day. Then they worked on the edits and writing the next paragraph, turned in a new draft, etc. (See what I mean? Large chunk of our time.)

Although self-conscious about it, I decided to provide a scripted template for the write-up for two reasons: to get them to wrap their brains around formal technical writing before requiring them to produce it themselves, and to allow them to focus on the portions I cared about most without completely doing away with the rest. The first paragraph is almost entirely scripted, with just a couple blanks to fill in. The other paragraphs allow much more freedom. I’m sure the template could be improved, so don’t judge! However, feedback is welcome.

Editable version of the assignment sheet: Ch. 12 Student Loans Project

Editable version of the template for the write up: Ch. 12 Studen Loans Write-up Template

*Unfortunately, my initial understanding of how student loan interest accrues was faulty. I was able to get this explanation from the bank managing my husband’s student loan:

Student loans are considered simple interest loans. Interest accrues daily on the outstanding principal balance.
The calculation used to determine the amount of interest that accrues per day is as follows:

Total Unpaid Principal Balance x Interest Rate, Divided by 365 or 366 (Days in a Year)

# Journal 10: Reflect on Semester One

Once again, we started a new semester with some reflection on what had gone before. This time I first asked them to choose five skills we’d worked on to rank in order of most understood to least understood. Then I asked them to write about the skills that were actually enjoyable to learn, and ended by having them reflect on their own actions and habits, good and bad.

Editable version: Beginning of Semester Journal

# Journal 9: Pre-final Vent

Journal 9 came right before the semester-one final, so I gave students a list of skills they’d been reviewing beside a continuum of fear adjectives and asked them to give vent to their states of mind. The value of the assignment lay in the combination of skill labeling, self-assessment, and end-of-semester therapy. There’s something to be said for letting students feel heard on a personal level in a math class.

Like several other journals I assigned this year, this one started as a fuzzy idea in my mind, then grew to have structure and support after I consulted with my assigned speech-language pathologist. Using the three sentence starters near the top, (N leaves me … because I …; One way I deal with this is …; I would rather N than…) the students first tried filling in blanks aloud in a class discussion. They seemed to enjoy that. Then I moved them to fleshing out their ideas in writing. I don’t think a single student wrote about the same topics they brought up in our discussion, which was interesting. Maybe they figured they’d already gotten those ones of their chests by the time they were writing.

*8/6/14. I listed all the skills as gerunds (solving…, remembering..., etc.), which fit perfectly into the “N leaves me …” sentence starter, but not into the “I would rather N than…” starter. The majority of my students failed to correct the grammar in the second case by changing solving to solve, and so on. In the future, especially when working with students with language learning disabilities, I’ll want to draw the class’ attention to the necessary change.

Editable version:Semester 1 Review Journal Therapy

# Journal 6: Rank the Solving Methods

Journal 6 asked students to rank the three quadratic solving methods they’d been working on: factoring, graphing, and the quadratic formula. This was the most heavily curriculum-related writing they’d had to do, which is probably why they struggled to describe the math with any specificity.

I liked having written evidence that preferences for solving methods differed, so I posted opposing excerpts in the classroom (for example, one student talked about preferring the factoring method, while another described how he hated that method the most).

Editable version: Ch. 7 Pre-test Rank Solving Methods Journal

# Journal 4: Mistakes

I’ve already talked here about this journal assignment , which I based on clips from Jo Baoler’s lesson about making mistakes, so I’ll just post the document. Here’s  an editable version: Ch. 5 Post Test Mistakes Journal

# Using Open Questions to Teach Simplest and Equivalent Forms

In September, while taking that Jo Boaler class I’ve mentioned, I finished a section about asking open questions that allow students to use different approaches and learn from each other. That’s a valuable idea, I thought. Unfortunately it’s just not compatible with my upcoming unit. Simplifying expressions is all about getting a single answer and there’s really just one way to do it.

The idea of asking open questions kept turning in my mind, though, until an application was suddenly obvious. I would keep the goal of having students write expressions in simplest form intact, but would add a dimension by having students also write expressions in non-simplest equivalent forms. This would allow students to follow their own thought processes and provide individualized responses, while also developing the concept of equivalence.

I wrote an expression in simplest form on the board, then asked students to write it in equivalent ways using

• Subtraction (the requirement of like terms for addition and subtraction was a worthwhile challenge)
• Multiplication
• Division
• Negative Exponents*

We used this activity during the simplifying unit, and returned to it during reviews of previous material. Some students understood the meaning of the activity quickly, while at least one didn’t perceive  it until just before the spring final: “Oh! We’re just writing things that will give us that first thing.” Bingo. Sometimes comprehension takes time. That’s why I’m in favor of continual review and intelligently designed cumulative finals rather than the take-a-test-and-forget-about-it model.

*I didn’t think to include negative exponents in the list until late in the year.

I’ve been loving the new things I’m doing this year. I should be posting more often, but hey.

I’ve started a pattern of spending the day after a test, before the tests have been graded, introducing a journal writing prompt for the kids to do for homework. After the test my kids took this week, I played two short clips from Jo Boaler’s video lesson about making mistakes in math class. The first clip described why mistakes are important to learning, how they can create two rounds of brain activity that don’t exist when problems are solved without mistakes. In the second clip Jo gave guidelines for teachers about how to get students to overcome their fear of mistakes. The writing prompt I gave students suggested that they

• Describe a time when they had made a mistake in my class
• Talk about how they feel and react after making a mistake in math
• Give suggestions for what the class could do to help them view mistakes differently

On average, the journals I received in response to this assignment were less insightful than I’d hoped. But one or two bright spots showed that my work on embracing mistakes as part of the learning process hadn’t been completely lost on the students.

Here are the two clips.

For some reason I can’t get the videos to start at the right point, so navigate to 0:46 on this one to see the bit I showed my class.

Start this one at 1:22.

# Non-equation Extension of Functions

My trig students are reviewing functions a bit right now. I used this piece of a chart showing average height by country that I found on Wikipedia to build the idea that functions and relations don’t only exist in the isolated world of equations and numbers. By assigning numerical values to information in a chart, you can turn just about any info into a mathematical function, or at least a relation.

We first assigned numbers to the countries of origin (1-11) so that we could have an input value instead of an input country. Using the male data seemed simpler because of it’s proximity to the list of countries,  and had the added benefit of the four N/A entries, so we could talk about whether the input 2 (Argentina) were really part of the domain if no output value exists for it.

We listed the domain and range, illustrated the mapping of input values onto output values, listed ordered pairs, and graphed them.

Viewing a chart as a function was a stretch for some (especially whenever I used the word input–“Input? What are we putting it into?”), but is a step toward breaking math out of isolation and realizing its contented existence beyond the classroom.

# Mathphrase

Even after my vocab-building epiphany, I was inclined to be overly forgiving of slow vocab development. Fortunately my school has speech-language pathologists on staff to collaborate with teachers, and my SLP was quick to direct me toward more exact standards.

For example, one of our favorite vocab-building activities was Math Catchphrase (later dubbed “Mathphrase”). I laminated little cards with mathematical terms on them (some were review words, others were new for current material). Then we played Catchphrase, but instead of reading the word from the disk you pass around, students grabbed their word from a turned-over stack of cards in the middle of the table. On each corner of the table was a piece of paper with the words “Skip Zone,” where students could put the words they had to skip so we could talk about them at the end of the round. After watching us play it once, my SLP saw that the game needed an obvious external incentive for the students to use mathematical descriptions of the words instead non-mathematical ones. So I spruced up the scoring system to reward mathematical descriptions, which made a big difference. Unlike regular Catchphrase, we would reshuffle the words and play again with the same stack for the sake of repetition and reinforcement.

For her end-of-year review, a student came up with her own vocab game (“Draw, Act, or Describe”) that I’m excited to throw into the mix this coming year to prevent overusing Catchphrase.

# Math Hatred Concept Map

I’m taking Jo Boaler’s open online course called, “How to Learn Math.” In the first session she has us watch short clips of college freshmen describing why they hate math, then asks us to sketch a concept map for the ideas we heard. Here’s mine.