Tag Archives: algebra

Standards-Based Grading in Fall 2018

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An Overview of My Semester

This semester, I’ll be teaching three different courses:

  • Pre-Calculus Mathematics (MATH 111), a course designed to review algebra and trigonometry for students who plan to take our (scientific) calculus sequence;
  • College Algebra (MATH 101), a course designed to cover algebra and function basics for students who will continue on either to MATH 111 (pre-calculus with trigonometry) or MATH 105 (business calculus); and
  • “Applications of Mathematics Across the Curriculum with Technology” (SMFT 516), a graduate-level course designed for in-service science & math teachers who are working toward an interdisciplinary M.Ed.

Due to enrollment challenges, my schedule for courses shifted in late July, so I spent a while during the summer trying to ditch my old plans for the semester and start over. Although I was planning to attend Mathfest in Denver, CO and give a talk in the session on #MasteryGrading based on my years of experience implementing standards-based grading in my courses, I must admit that before my trip I had no plans to use SBG in any of my courses this semester.

But then… UGH!!! INSPIRATION FROM PEOPLE. AND TOO MANY GOOD IDEAS.

So I attended every talk in the #MasteryGrading session at Mathfest. And wow, I got a ton of great ideas from all of the talks [stay tuned for future blog post] and, on a personal level, I really enjoyed our conversations, meals, and hang-outs outside of the session itself. (Thanks, y’all!)

Unfortunately a couple of days into Mathfest I realized I just couldn’t go back to traditional grading, so I threw out all my traditional plans for the semester and committed to myself that I would implement SBG/SBSG/MasteryGrading in 100% of my courses this semester.

Did I mention that I got home from Mathfest only 15 days in advance of my semester start?

The Nuts and Bolts of Fall 2018: SBG PreCalculus and SBG College Algebra

Rachel Weir, of Allegheny College, is maintaining a repository of course documents for secondary Mathematics courses that are using Standards-based grading, Specifications-based grading, or Mastery-Based Grading: Rachel’s SBG Repository

Both my Pre-Calculus and College Algebra courses are using the exact same setup. It’s very similar to Tom Mahoney ‘s (@MathProfTom) approach in his College Algebra courses. Here is the basic setup:

  1. I have written 25 standards for each course.
  2. Every time a student completes a problem on a standard, I will assess the solution using a “SGN Rubric” (see below). This assigns either 0 points, 1 point, or 2 points to each attempt.
  3. A student’s score on a standard is the average of their best two attempts.
  4. A student earns total points out of 50 possible (25 standards*2 points max). Together with work in an online homework system, this converts to a usual letter grade*.

For example, for any given standard, I will track a student’s progress as something like “0,1,2,1,1,1,0,2” and this student will earn a 2. After two perfectly correct solutions, the student isn’t required to answer problems on that topic again.

*My department requires a departmental-wide final exam that is graded using a partial credit, percentage system, and this exam must be worth at least 25% of each student’s course grade. So the actual grade computation is (75% performance on standards)+(25% final exam performance).

Links to Possibly Useful Things

Here are some links that I’ve freely distributed to my students. Perhaps reading them will shine some light on how I explained this system to them. Also, there are more details about the “SGN Rubric” I mentioned above and explanation about online homework & how it fits in.

Things To Do Later

I haven’t mentioned that third class (“Applications of Mathematics Across the Curriculum with Technology”). It runs double speed for half the time, during our Express II semester, and it doesn’t start until October. I want this course to be a project-based course, so I’m going to figure out some way to introduce specifications grading into my design. Robert Talbert (@RobertTalbert) has written extensively about his use of specs-grading and it’s my plan to steal as many ideas from his MTH 350 F18 Syllabus as I can. Our courses are very different, but he has so many clever ideas for his course skeleton. Once I write my syllabus, I’ll tell you about it.

Modeling Fun with Paper Fish

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Kate Owens, 02/2016

Kate Owens, 02/2016

Back in early February, as part of my ongoing work with the Math & Science Partnership, I led a Saturday professional development workshop for STEM teachers on “Proportion, Decimals, and Percents (oh my!).” There were two major projects we worked on that day. First, I split the teachers into teams of two or three and they read over some “Always, Sometimes, Never” statements. Fawn Nguyen’s blog post has some great ideas to get you started on those. Second, we simulated determining a wildlife population. Since this is something I hadn’t seen blogged about before, I thought I’d tell you about how it worked.

I found the idea for this in a book called “Mathematical Modeling for the Secondary School Curriculum.” It’s based on an article called “Estimating the Size of Wildlife Populations” that appeared in the NCTM’s Mathematics Teacher back in 1981*. Here’s how it works. Suppose you have some closed ecosystem that has a population of animals — maybe a large lake containing a population of a certain species of fish. What if we want to know how many fish are in our lake? Can you think of ways we might approximate the number of fish?

Here are some ideas that might spring to mind:

  • If we knew something about the social personality of the fish — for instance, maybe they are really independent and territorial and don’t like hanging out together — then we might know that they prefer to have at least X cubic meters of space to themselves. If we knew how big the lake was, then this could give us a rough count on how many fish there are. Problem: Knowing how big a lake is, in terms of volume, can be tricky. The bottom of the lake might not be flat. The amount of water varies based on temperature and rainfall. And what if we don’t know if our fish are social swimmers or solo swimmers?
  • We could rope off (fence off? net off?) a portion of the lake and count how many fish are in our section. If we knew we’d roped off exactly 10% of the lake, maybe we could use this information to estimate the total number of fish. Unfortunately, this is also difficult. First, we don’t know the fish are uniformly distributed around the lake. Maybe we roped off a portion of the lake that’s very rich in food source so we have many more fish than we should. Second, it’s tough to know if we’ve gotten exactly 10% of the lake or not. (How do you measure the volume of a lake, anyway? I’m sure there’s some way to do this, but I have no idea how.)

You may have thought of some other ways, too. Leave them in the comments. Here’s the way proposed in the NCTM article. It’s known as a capture-recapture estimate. Let F represent the number of fish in our lake. First, we capture a large number N of fish and tag them in a way that isn’t harmful; then we toss them back. We wait a while. Once the fish have had a chance to do their fishy things, we go back to the lake. We then capture x fish — some are tagged(T for Tagged), some are not. Assuming the fish are randomly dispersed throughout the lake, we might conclude that the number tagged in our sample is proportional to the number of tagged in the entire lake: N/F = T/x.

For a quick example, suppose we capture and tag 1200 fish. When we return to the lake, we re-capture 200 fish and we find that 30 of them are tagged. Assuming that the number tagged (30) in our sample (200) is roughly proportional to the number tagged in the lake (1200), we conclude that 30/200=1200/F so F=8,000.

What could go wrong? Well, maybe our sample isn’t very indicative of the population. We throw back all of the fish and then take another sample of roughly the same size. If we take several different samples, we can use the additional information from further samples to get a better estimate of the fish population. (I’m not going to go into all of the statistics at work here.)

Modeling the Fish Population

I gave each group of teachers a box. A shoe box would work. Inside each box were about 200 squares of paper. I didn’t count the squares as I put them in, and I didn’t want any two boxes to have precisely the same number. Having ~200 isn’t necessary — you just want enough people can’t do a fast eyeball estimate, but not too many because eventually you’ll want to count them.

One teacher “went fishing” and “tagged” a handful of fish (a dozen or so) PDP-fishby marking those squares with a signature, symbol, smiley face, whatever. The fish were returned to the lake before they suffocated. The box was shaken up. Another teacher then took a sample of size larger than the tagged number — something along the lines of 20-25 fish, give or take. The number of tagged fish in each sample was counted. Fish were returned to the pond, the box was shaken, and like it says on your shampoo bottle, “Lather, rinse, repeat.” Assuming the captured sample was the same size, after taking 10 samples, we averaged the number of tagged fish. Using proportions, we found an estimate for the total number of fish in the pond. Lastly, each team counted the actual number of fish in their pond to see how close they were. Most groups were pretty close. As an extension, we discussed how we might modify the method if more than one species of fish were in the pond.

(I saved the boxes. If I do this experiment again, I need to remember to make sure there are lots of squares of paper. Students were easily “fishing” for 30+ fish at a time, and so sometimes they’d end up capturing all their tagged fish.)

The teachers enjoyed this activity & I hope they’ll try something similar with their own students. We had a lot of great conversations about ecology and how our method could be extended, what flaws it might have, and so on!

*Knill, George. “Estimating the Size of Wildlife Populations.” Mathematics Teacher 74 (October 1981): 548′ 571.

On Algebra

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There have been a whole flock of article recently addressing the question, “Should we teach algebra to all high school students?” It started (I think, anyway) with Andrew Hacker’s Op-Ed post, “Is Algebra Necessary?” in a recent issue of the New York Times. His conclusion: No, algebra isn’t necessary. A careful reader ought to question, “What does Hacker mean by algebra?” and “Do we want an emeritus political science professor to make decisions about the mathematical education of the masses, given that there are so many people whose entire careers are dedicated to mathematics, the research of mathematics teaching and learning, and being mathematics educators?” But today I wasn’t planning on addressing those questions or Hacker’s article. (See Daniel Willingham’s response, “Yes, algebra is necessary” if you’re interested.)

Instead, there are several important issues that I think are worth pondering whenever anyone starts talking about the necessity of algebra.

1. What do they mean by algebra?

Out of curiosity, I asked WolframAlpha to tell me about “algebra”. It gave a lot of responses (but no definition). It did, however, provide a clear distinction that algebra is something more than equation solving. Yet in Schank’s first paragraph, he seems to conflate all of algebra with the quadratic formula:

“Whenever I meet anyone who wants to talk about education, I immediately ask them to tell me the quadratic equation. Almost no one ever can. (Even the former chairman of the College Board doesn’t know it). Yet, we all seem to believe that everyone must learn algebra.”

I’ll skip over a discussion pointing out that he means the quadratic formula but wrote “the quadratic equation” (as if there’s only one).

In any case, very few people have given the word “algebra” a good-enough definition, from my viewpoint. When I put on my research mathematician hat, I like thinking about universal algebra, which is somehow even more broad and amorphous (and beautiful) than the above definition could convey.  (I am even a published, theorem-proving universal algebraist.) I define algebra like this:

Algebra. Noun. The branch of mathematics that deals with the study of structure.

Yes, that includes studying equations. But it also includes wilder animals like  finite fields or nonfinitely axiomatizable equational theories and the varieties they generate!

2. Who is the “they” and who is the “us”?

Schank also asks,

“Are mathematicians the best thinkers you know? I know plenty of them who can’t handle their own lives very well.”

It turns out that, despite lots of evidence to the contrary, mathematicians are people. As in, real people with real lives and real feelings and real kids and real cats and real hobbies. And, sometimes, real problems. I don’t know why this is news. Surely no one would suggest that we ought not listen to music (or teach music in schools) because some musicians have had difficult lives. The people behind the subject is what makes it compelling. If only robots did math, I’d probably be less interested in math. [And more interested in robots.]

How do mathematicians think? Wow, that’s a fabulous question. Look, no mathematician I know claims that we think “the best.” That’s a “they versus us” distinction if I ever heard one. However, many of us do claim that we think differently.

To believe anything, a mathematician requires a proof.

This is different from every other academic discipline. Mathematicians have a very strict code for how we think about problems. Intuition is never enough.

3. Who taught these people mathematics?!

Moving toward his conclusion, Schank writes,

“You can live a productive and happy life without knowing anything about macroeconomics or trigonometry but you can’t function very well at all if you can’t make an accurate prediction or describe situations, or diagnose a problem, or evaluate a situation, person or object.”

(It humors me that the end of the article talks about making “sensible political choices,” but here it turns out we don’t need to understand any macroeconomics. Say what?)

I wonder who taught these people algebra! None of my students will survive my courses unless they are able to demonstrate that they can use the mathematical content knowledge from my class and apply it to real-world problems about situations that involve optimization or diagnostics or evaluations or predictions. Who are these [potentially imagined] math teachers who teach nothing but endless factoring of polynomials without any motivation?

4. What’s a better question to ask?

Schank and I agree that “[t]he ability to reason from evidence really matters in life.” He thinks algebra doesn’t help develop the skills to do this — I disagree hugely. Algebra can help develop this skill. Does every math educator teach it perfectly? No. Could all of us math educators do a better job? Yes.

My goal as an educator of mathematics is to converge asymptotically on being amazing at my job. Could teaching algebra help students understand logic, reason, and critical thinking? Absolutely. Does it always? Maybe not. But that is not a reason to say, “Don’t teach algebra.” We should say, “We need to teach algebra better.”

And, I promise, I am trying to do just that — along with thousands of my colleagues around the world.