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What Teachers Make

Formative Assessing to the Extreme

For the past couple of weeks, I have been taking How Math Must Assess to an extreme, literally checking every problem every student does in class, providing constructive feedback to each student who needs it, and placing a check next to their names if they demonstrate to me that they can "do" whatever it was they were doing on that day. And it really is easy.

For example, this past week was a short week, having Monday off. My goal for this week was to have students solve right triangles. I really began last week by reviewing the Pythagorean Theorem the Friday before after a very short quiz, checking off on every student who demonstrated to me in class that they could actually find any side of a right triangle when given any two of them, providing constructive feedback for those students who needed it and checking them off after they could demonstrate to me that they could do it. Tuesday, I introduced the three trig ratios and how to use them to find any side of a triangle. Wednesday, we went from sides to angles. Thursday, we put everything together.

What I do is this: After some sort of a "mini-lesson" of fifteen minutes or less, I provide the remaining thirty minutes or so for students to apply the ideas learned during the mini-lesson on a problem set containing somewhere around fifteen to twenty problems, broken up into groups of four or five problems. I do NOT circulate around the room. I pull up my chair and sit next to an empty student desk somewhere in the room, and I ask my students to get up, walk to where I am sitting, and check their work with me after each group of problems. This seems to provide my students just enough movement-with-a-purpose that they can work rather diligently for the entire class.

When a student comes up to me, I can quickly scan their work and answers and provide specific feedback ("If you are going to use this angle in your calculations, then this side is the adjacent side"). If, after looking over their work for a group of problems, I am satisfied that a student can do things correctly, I place a check next to their name. For some students, this may not occur until after they have received feedback from me and re-worked some of the problems. For other students, if it looks like some sort of a calculation error on their part, I may send them back to their seats and have them shout out the answer to me without making the trip back up and standing around my desk ("It looks like you divided 4.3 by the tan(5) instead of the tan(55). Go back and double check that you end up with something around 4, and shout out your answer to me when you get it.").

I have 28 students in three of my four classes that I have been doing this with (the fourth class has 13 kids). I may have a group of four or five students waiting around me to have their work checked, but I ask them to listen carefully to any feedback given to another, because it may apply to them and their work.

My hope is that by the end of the week, every student will have demonstrated that they can meet the objectives for the week before we ever have a quiz. The results from last week? Fifty-eight students in the four classes combined could solve the right triangles perfectly. Only eleven students out of the four class could not solve the right triangles. Of these eleven, nine of them did not demonstrate to me during the week that they could do the work anyway, so this was not a surprise.

Anyway, I plan on continuing my extreme formative assessment this week as work with special right triangles and their trig ratios. We'll see how it goes.

Median Grading (Part 2)

Two other short articles about median grading: One Example of Median Grading Another Way for Teachers to Compute Student Grades and Teachers Need to Look at Median Grading: Evaluation and Instruction Suffer When Grades are Averaged. 

If you look through the author's other articles, there are some thought-provoking reads.

What if I used the Median to Calculate the Grade?

I have read a couple of posts on different blogs lately about grading, standards-based grading, and assessment. In particular, Why I Hate Grades, Mathematically Speaking at yofx and Another Standards-Based Grading Question at A Math Teacher's Notebook. Thinking about a system like Dan Meyer's found in his How Math Must Assess, turning student mastery records into end-of-quarter grades using the mean as a measure of central tendency seems difficult. After reading Success for All: The Median is the Key, is the median a viable option?

Euclid and his Modern Rivals

I picked this book up on a lark the other day, and I find myself wishing I had a copy of Euclid and some of the other geometry texts referenced in this Lewis Carroll (published under his real name, Charles Lutwidge Dodgson) gem. Reminds me a little bit of Imre Lakatos' Proofs and Refutations.

100 Visitors

One hundred visits in just over one month (started this puppy on December 13, 2009).

I am certainly glad that folks have found my blog. It is still evolving...more than anything, I wanted to start a blog as a means of improving my writing. I want to write and publish more than I have (I have written or co-written a couple of things ranging from articles for my state journal to the Mathematics Teacher to the American Mathematical Monthly) but writing is difficult for me. Hopefully, by regularly blogging, it will make the writing process easier.

That said, thanks to the folk who have contributed to the 100 visits!

More of My Mental Rube Goldberg Machine

It once again starts with a trip down Farnham Street and ends up with my head spinning.

In the Farnham Street post Can You Measure?, I was led to the article Shape of glass and amount of alcohol poured: comparative study of effect of practice and concentration, with the objective to 

To determine whether people pour different amounts into short, wide glasses than into tall, slender ones

leading to the conclusion

To avoid overpouring, use tall, narrow glasses or ones on which the alcohol level is premarked. To avoid underestimating the amount of alcohol consumed, studies using self reports of standard drinks should ask about the shape of the glass.

which reminded me of two posts over at dy/dan, Don’t Forget Answers, Iteration and What Can You Do With This: Club Soda. 

I was also led to the article Downsize in 3D, Supersize in 1D: Effects of the Dimensionality of Package and Portion Size Changes on Size Estimations, Consumption, and Quantity Discount Expectations, which found that

changes in size appear smaller when products change in all three dimensions (height, width, and length) than when they change in only one dimension

leading to the conclusions

consumers expect (and marketers offer) steeper quantity discounts when packages and portions are supersized in 3D than when they are supersized in 1D; consumers pour more product into and out of conical containers (in which volume changes in 3D) than cylindrical containers (in which volume changes in 1D); and consumers are more likely to supersize and less likely to downsize when package and portion sizes change in 1D than when they change in 3D.

which reminded me of the blog post Price Realization Through Creative Package Sizes over at Iterative Path.

Not sure what this all means. I would like to find a way to use all of this in my classes some time soon.

My Mental Rube Goldberg Machine

I was over at Farnham Street reading Starbucks and the Rule of Three, which led me to William Poundstone's blog Priceless at Psychology Today his post Decoding Fast-Food Menus, which led me to the video below, which reminded me of Dan Meyer's recent post How Do You Turn Something Interesting Into Something Challenging? and his series of What Can You Do With This? posts. 

Not that I am ready to turn any of this into something challenging, nor am I considering a WCYDWT? idea, but there is something here.

Too Much Time on Your Hands?

Now, if you would only memorize the digits...

Pi buster: French software whiz claims record

Interesting Fraction Division

After reading Richard Skemp’s Relational Understanding and Instrumental Understanding at the Republic of Mathematics it got me to thinking about something that happened in class this past week.

I teach a class once a week at a local community college close to my school. I like it...it is a nice change, with an amazing mix of students; some, just out of high school, and others being older adults looking to go back to college or to enter college for the first time. I teach a sequence of three courses - Preparatory Mathematics, Introduction to Algebra (part 1), and Introduction to Algebra (part 2) - none of which count towards any of my student's degree plans because they are not considered to be college-level mathematics.


Teaching these classes are very rewarding. I had an older adult who stood outside the classroom door for five minutes on the first day of class last quarter, afraid to come in because she thought she was going to fail the class. She went on to be one of my better students. To see how she (and the rest) grow to realize all the mathematics they know and can do is special. I typically have about 75% of the students from the Prep Math course take all three classes in the sequence with me.

On a side note...it is funny is to see their reaction to how I let them retake parts of quizzes and the like during my office hour before class (actually, it is really funny to see their reaction to WolframAlpha or to GeoGebra). I explain how some students will get it sooner than others, and how I want their grades to be an accurate assessment of their most current understanding of the content. Everyone in the class has gone through high school, they know the game, so this idea is a shock to them. About halfway through the first quarter, they get it, and start stopping in during my office hour. Of course, there are some institutional constraints that are much different than high school, but as long as I can get everything done that is in the syllabus, I am cool.


Something came up this past week when reviewing fraction operations, in particular, division. We discuss the usual finding-of-common-denominators when adding and subtracting, and how you can, but you do not need to, find a common denominator when multiplying or dividing. Many of the students choose to because it is one less thing to remember, and I am ok with that. I am not going to allow fractions to be a deal breaker for them. Show me you can add, subtract, multiply and divide some everyday fractions, that is good. 


When it comes to division, we discuss the normal sorts of mnemonic devices they remember from high school. Same, change, flip. "We do not wonder why, we just flip and multiply." How about THAT for a mnemonic device! The usual question arises: "Why can't you just divide straight across in the same way you multiply straight across?" Since they ask, we talk about it. You can divide fractions by dividing the numerators and denominators straight across, but this method will sometimes leave you with a fraction in the numerator and a fraction in the denominator.

Not a big deal. This just means that you will need to do some additional work to change this fraction into one with integers in the numerator and denominator by rewriting the fractions with common denominators

which of course adds a layer of fraction sophistication, such as multiplying the numerator and denominator by 20.


This past week, someone who liked to find common denominators for multiplication and division problems, noticed that when you found a common denominator first, then divided straight across, you could avoid the messy fraction with fractions in the numerator and denominator.

Thought this was pretty cool. I had never thought about fraction division in this manner, probably because I never found common denominators when I divided. I just always flipped and multiplied!

Things That Make Me Go "Hmmm..."

From Dan Meyer's Blog How Do You Turn Something Interesting Into Something Challenging? and the idea of lowering mathematics into the discussion s l o w l y. First thought that jumped into my head? I suck at teaching math.

From Kate Nowak's Blog Soldiers in the War on Innumeracy and the link to the Verizon Math video. I watched this video with my 8th grade son last night, and I think he wants to join this War.

The closest I have come to this nirvana is using this video when discussing the Prisoner's Dilemma and other Game Theory ideas with my Mathematical Games class last semester.

Fermat Point

Sticking with the name of my blog, the Fermat Point was the first point of concurrency to be spotlighted. Not sure why I chose this point...I probably should have followed Clark Kimberling's Encyclopedia of Triangle Centers and chosen the Incenter, which is center X(1) in the ETC. The Gergonne Point is the next to be spotlighted, and it is closely related to the Incenter, so I think that could count.

The Fermat point, also known as the Torricelli point, is the point with the property that the total distance from the three vertices of the triangle to the point is as small as possible. The point takes its name from Fermat, who proposed this problem to Torricelli. More on the Fermat Point can be found at Cut-the-Knot. I use this with my geometry students after we study inscribed angles.

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