Ed School Screening and Beyond, Dept.

I applied to George Mason School of Education in the fall of 2005. They had a special program for people in the workforce called the “Career Changer” program. It was aimed at people like me who wanted to get into teaching after having been out in the working world. In my case, I had been out in the working world for over 30 years, and was preparing for a career of teaching after I retired which would be five years from then.

In an effort to make it look like it was a special hard-to-get-in program, the school held a meeting in which the applicants had to go through a series of interviews, and then prepare a writing sample. At the introductory session, a woman addressed the candidates, as we were called and said in her opening remarks that “School is not your father’s classroom anymore.” Holding up an index finger, she then declared what it was: “Inquiry-based!” I might have been one of the few in the crowd who knew what those words meant. I knew that many of the people in that room would be swayed to the “not your father’s classroom” standard.

She went on, mostly about what to expect in teaching and then made a plea for getting a masters in education. “Research shows that teachers who only have a teaching credential tend to leave the profession after three years, but those with masters degrees stay the course.” I resolved not to do that, and don’t regret it.

One of the events of the evening was being interviewed by various ed school professors who asked us questions such as “What do you see yourself doing in five years?” which were probably designed to see how well we could bullshit and sound sincere. The capstone event was having to prepare a writing sample. We were sent to a room with computers and told to write an essay on a topic whose theme I can’t recall, but it was some broadly bland theme like “What is the Value of an Education?” and we had to come up with 1500 words. I think I wrote something along the lines of “Lack of education results in falling prey to things like “inquiry-based classrooms are better than direct instruction”– but phrased a bit more diplomatically. I even included references.

A few weeks later I received my acceptance letter which reminded me of those promos one receives in the mail notifying you that you may have already won one million dollars in some sweepstakes competition. I spent the next four years taking one class per semester at night, and finishing up my student teaching in California, after having secured a tentative agreement with George Mason that they would cooperate with Cal Poly–which they tried to get out of. I hadn’t counted on the people who had been so agreeable to the idea leaving the college or taking new positions within it. My new advisor reluctantly went along with it because I had the foresight to get the agreement in writing.

About the only thing of value I got out of ed school was hearing the professors’ stories of their days as teachers–it was a lot more useful than the textbooks we had to read, or the 3 hour night-time sessions. One class on “Methods of Math Teaching” consisted of the teacher stating a particular category (like “attributes of effective teaching”) and asking the class to come up with ideas. She would write down the ideas, filling the white-board with a list, and then would proclaim “Good list!” There were other things that filled the three hours, such as students reporting out on an article that was assigned. I recall reporting out on an article on doing away with grades, and using “standards-based” descriptions of students’ performance. The strange thing about it was that in reporting the article, I was presenting the side of the author and essentially selling the idea to my classmates, who–I’m pleased to say–were having none of it. Maybe I meant for that to happen–yeah, it was “intentional”. (The edu-word “intentional” hadn’t yet achieved its current status of popularity at that time.)

In another class (“Literacy in the Content Area”) many of the students were already student-teaching. I worked full time so I couldn’t do that. One assignment was for students to report out on a technique they had used to teach a particular topic. A student who taught social studies talked about how he taught the unit on civil rights, and in particular the Freedom Riders. He recounted how the civil rights workers had their bus torched in Mississippi, but the next day, got on another bus and continued their brave journey. It was indeed an inspiring story, but now his assignment (this was for his class of eighth graders) was to have the class pretend they were Freedom Riders, and their bus had just been torched. Those who chose to continue the ride were to stand on one side of the room; those who chose not to, would be on the other side. Our class now did this activity.

In deciding whether I would stay on the bus or decline, I considered my own situation: I have a wife and child, so that if I were to die, they would be left in the lurch. So I was one of two people who declined and stood on one side of the room while the rest of the class stood on the other. I felt guilty and ashamed; I could only imagine how eighth graders would feel doing such an activity. I thought the activity was inappropriate and that one could simply ask students to think about what they would do if they were a Freedom Rider and leave it at that. No need to put them at risk for being called “racist”, although now-a-days this has probably become de rigueur in classrooms. Maybe this student was way ahead of his time.

All in all, the most useful skill I took with me from ed school was how to make it look like I was going along with the party line, while doing what I felt was right for the students. And fortunately I retired just before having to participate in PD devoted to checking one’s privilege and admitting one’s white fragility became education’s shiny new thing.

Traditional Math (12) Translating from Words into Algebraic Expressions (7th Grade)

This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. (Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!)

1. Introduction/Opening Monologue

Translating English into math is an important skill and is basic to solving all word problems. They have done this to a limited degree using arithmetic methods which I use in my opening monologue to the day’s lesson. I start off asking them to tell me how to write the numerical equation for various statements.

I quickly give a worked example to ward off students giving me the answer to the problem, though I can assure you that no matter how many warnings and instructions you give, that will occur: “The cost of ten yo-yos if each costs three dollars.”  What I’m looking for is the numerical equation, not just the answer; in this case it’s 10 x 3 = 30.

Now I give the statements, telling students to write it down in their notebooks or on whiteboards:

“The number of students on three buses if each bus holds twenty two students”  Answer:  3 x 22 = 66

“The amount of money Nina earned if she mowed the lawn for $15 and walked the dog for $4”.  Answer: $15 + $4 = $19

“The number of students in each group if fifteen students are divided into five equal groups”  Answer: 15 ÷ 5 = 3 or  15/5 = 3

The process of writing statements numerically is then extended to expressing statement in algebraic terms in which variables are used.

2. Translating into algebra

Warm-Ups. For this lesson I typically include warm-ups that provide a preview to what we will be doing, and serve as a segue to the day’s lesson. Examples include:

  1. Write the numerical expression for 5 more than 10. (Answer: 10+5 or 5 + 10)
  2. Write the numerical expressions for 5 less than 10. (Answer 10 – 5)
  3. Write as an expression: Seven times x (Answer: 7x)

Defining the Variable. Variables will represent unknown quantities. If I say “some number”,  since we don’t know what that number is, it can be represented by a letter. Taking a problem from one of the warm-ups, I’ll point out that if I say “five more than ten” we write that as 5 + 10. I’ll ask: “Suppose I say ‘five more than some number’.  How would I write that?” There is general consensus reached quickly that it is 5 + x, which can also be written as x + 5.

I give them numerical forms first and then extend that to variable.  “Five times a number” (5x)

“A number divided by 3”.  I’m looking for x/3, but if I get x ÷ 3, I accept it, and quickly point out that in algebra we write it in fractional form.

Now I start to use unknown quantities that name specific things, like “Two miles more than an athlete ran.”  If there is a stunned silence, I’ll ask what the unknown quantity is, and they’ll pick up that the answer is 2 + x.

“Be careful on this next one,” I’ll warn. “I’m betting that at least people will get this wrong.”  The problem: “5 less than some number.”  Usually many people will say 5 – x.  This particular error is the gift that keeps on giving. Some students will repeat the mistake through the year; so it is good to keep repeating such problems.

I advise at this point that if a statement is confusing, see how to write it using numbers, and I refer to one of the warm-up problems. For example “5 less than 10 is obviously 10-5.” The bottom line answer is five is being subtracted from a number.

It is helpful to mix in other terms for “less than”; for example “5 fewer than some number” and “some number decreased by 5” are equalivalent to 5 less than some number. At this point it is important to make the distinction between “less than ” and “is less than”.  The phrase “5 is less than 10” is written 5 <10, and similarly “5 is less than some number” is written 5 < x.

After working with addition and subtraction, I turn to multiplication and division, and ramp the problems up so they are doing both. Starting with “Three times a number”, and “The cost of some number of games of bowling at $4 per game, for example, and ending with “Four less than two times a number.”

More Problems.  After the students are fairly comfortable with the initial translations, I give more “story-oriented” problems, that increase with difficulty.

Here are some taken from Brown, et al (2000) that I like to use, and which I also include in homework worksheets. Students are to provide the expression in terms of the variable:

1, The Tigers had twice as many hits as the Yanks. If x = the number of hits by the Yanks, then ____ = number of hits by the Tigers.  (Answer: 2x)

2. The length of a rectangle is four times the width. If x = the width, then ____ = the length. (Answer: 4x)

3.  Mac is x years old. How old will he be next year? (x+1)

(This particular problem causes students difficulty because although they have been writing expressions in terms of a variable, some think that this problem is asking for a number.  I remind them that they are to express Mac’s age in terms of x.)

4. Trish is t years old. How old was she 7 years ago? (Answer t-7)

5. Karen will be m years old next year. How old is she this year? (Answer: m-1)

6. Pete worked 4 hours more than Quinn. Quinn worked 2 hours more than Rob.  If x= the number of hours Rob worked, then  ___ = the number of hours Quinn worked and ___ = the number of hours Pete worked.  (Answers: Quinn’s hours = x + 2;  Pete’s hours: x + 6

 

Reference: Brown, Richard G., Smith, G.D., Dolciani, M.P.; (2000). Basic Algebra; McDougal Littell, Illinois.

 

Traditional Math (11) Algebraic Expressions (7th grade)

This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. (Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!)

  1. Introduction

Students have had some exposure to solving equations in their earlier courses, having to solve problems such as 3 +n = 10, and 13 – n = 5. These are solved using the arithmetic properties of numbers using the relationships known as “number families” or “number bonds”.  These concepts view a number addition equation in three ways. For example 8 + 4 = 12 lends itself to two other related equations, namely 12 – 4 = 8 and 12 -8 = 4.

Faced with the problem 3 + n = 10, the number family approach teaches students that n can be expressed as 10 – 3, and therefore n = 7.  For a problem like 13 – n = 5, the student know the number family 5 + 8 = 13, and therefore 13 – 8 = 5. He can also approach the problem by exchanging the n and the 5 to obtain 13 – 5 = n.

In seventh grade, students are given an introduction to the basics of algebraic expressions and learn to solve simple equations using the tools of algebra. This represents a different approach than they are used to and is a much more powerful method that allows them to solve more complex equations and provides an approach for solving word problems.

I like to start the unit by giving them a problem, and prefacing it with the following statement: “I’m going to give you a problem that you will think you know the answer to and you will probably be wrong.”  This serves as a challenge and a dare and also defuses the fear of making a mistake because they will want to prove me wrong. The problem is: “John and his sister have $110 between them. John has $100 more than his sister. How much do each of them have?”

Almost instantly hands are raised and students will call out confidently: “John has $100 and his sister has $10.”

I say that that is incorrect, because the problem says John has $100 more than his sister. If she has $10 how much will John have?”

They quickly figure out that he would have $110. “And what is the sum of $110 and $10?”  Seeing their error, some of the students then resort to a “guess and check” procedure, trying various combinations.  Someone will inevitably say “John has $90.”  I respond that if so, then his sister has $100 less. What is $100 less than $90?”  I cross my fingers that they remember how to work with negative numbers and one of the braver students will volunteer that it is -$10. Since you can’t possess a negative amount, we know that is wrong.  Eventually after enough guessing and checking they come up with John has $105 and his sister has $5.

While “guess and check” is a strategy that can solve problems, I point out that it took some back and forth before they came up with the right answer. “There is a way to get that answer on the first try,” I announce. That way, of course, is algebra. I tell them that they will learn to solve this problem and others using algebra in this unit.

  1. Writing and Evaluating Algebraic Expressions

Warm-Ups. Typical Warm-Ups for this lesson should incorporate past concepts.

  1. A 5 foot length of ribbon is cut into 2 ½ inch strips. How many strips are there? (Answer: 5 ft = 60 in, so 60 ÷ 2 ½ = 60 x 5/2 = 24 strips)
  2. -2 ½ x 3 2/5 (Answer: -5/2 x 17/5 =-17/2 = -8 ½
  3. (8/3)/(5/6) (Answer: 8/3 ÷ 5/6 = 8/3 x 6/5 = 16/5 = 3 1/5
  4. If you lose $2 every week for four weeks in a row, what is your loss after 4 weeks, and the fifth week you make $7, do you have a net gain or loss, and by how much? (Answer: -2 x 4 = -8; -8 + 7 = -1; a loss of $1.)
  5. What is 5 times n if n = -3? (Answer: 5 x -3 = -15).

Students have had experience using letters to represent numbers as discussed above.  Now we take it further with the goal of being able to represent English in terms of algebraic expressions. To do this students need to know the general rules of variables, and what a variable is.

A letter used to represent a number is called a variable. I liken it to a fill in the blank.  The sentence “I have __ apples” can have different meanings depending on the number that is used to fill the blank.  “I have x apples” does the same thing. The variable x in this case represents a changing—a fill in the blank type—number. The formal definition that I give to students is:

A variable is a letter or symbol used to represent an unknown value. Any letter can be used as a variable.

Addition and Subtraction. To be able to work with variables it is helpful for students to plug numbers in to various expressions that have variables, and to calculate the value of that expression. We start the process by looking at addition and subtraction.  I ask students if I have some unknown amount of apples, I can represent that by a variable.  I then ask students if I let n represent the unknown amount of apples and then want to represent 3 more than that amount, how would I write it?

I point to the definition of variable which states that the variable represents an unknown value. I will tell them: “In algebra, when we don’t know what the value is of a quantity, we represent it with a letter.”

Students are generally slow to respond to this but eventually someone will say the correct answer of n + 3.

I give more examples of this nature: “I have an unknown amount of apples and I give 5 away. How do we represent this?  x -5.

I will have them look at the first example of n + 3, and ask how many apples does the expression represent if n equals 5. I continue with other numbers: 100, 2,538, etc.  In the second example of x – 5, I might ask “What is the number of apples if x equals 10, 27, 5?” and so forth.

Now I might ask: “If I have an unknown amount of apples and call that amount m, and I get more apples of the same amount; how would I write this?”  They may hesitate a bit, but I am after m + m.

I now want to take it out of variables representing objects and just letting the variables represent “a number”.  I have an unknown number; I’ll ask how do I represent an unknown number? By now they’re in the rhythm of the questioning and will call out letters; usually x, but I want them to know x isn’t the only one they can use.  I’ll now ask I want to represent 5 more than that unknown amount. Next, with the x + 5 written on the board, I ask what is the value if x equals 4; then 3,  then 0,  then -5, -10 and so forth.

Finally, I’ll ask what happens if I have an unknown number and I add another different unknown number to it. How would I represent that?  I tell them to write it in their notebooks, or on a mini-whiteboard. I’m looking for two different variables added: a +b, x + y, and so on.

Like Terms. At this point I introduce some additional vocabulary: Term, like terms, and numerical coefficient.

When addition or subtraction signs separate an algebraic expression into parts, each part is a term. For example a + b consists of two terms, a and b.  Suppose I had 2a + 3b. Then 2a and 3b are terms.   The numerical part of a term that contains a variable is called the numerical coefficient of the variable. I will ask what the numerical coefficient is of 5x, of -24y.

If two terms have the same variable, or combination of variables, they are called “like terms”. I provide examples such as 2x and 34x, 8ab and 5ab.  Because the terms are the same, they can be added, just like we added x + x previously. This tends to be confusing at first, so I will liken it to “like objects”.  For example, if I have 2 apples and then get 3 more, I am adding 2 apples + 3 apples for a total of 5 apples. The variables can be thought of similarly.  So 3x – 2x + 5x, is the same as adding the numerical coefficients—that is 3 – 2 + 5—and adding the variable afterward: 10x. The summary statement I give them is:

To combine like terms that have variables, add or subtract the coefficients.

Multiplication. With the above as an introduction, I move on to multiplication. I ask students if I have 3 boxes and each box contains 5 apples, how would I calculate the total amount of apples?  They immediately know it is 3 x 5, but I write it as 5 + 5 + 5, and then next to it I write 3 x 5.  If each box has 7 apples, then similarly we have 7 + 7 + 7, which is written 3 x 7.

What if I don’t know the number of apples in each box, I’ll ask. How would I represent the total number of apples?

If x is the variable, then 3 boxes with an unknown amount of apples in each one can be represented by x + x + x.  How can we write this as a multiplication statement like before?  In algebra we represent it as 3x, meaning 3 times x.

After stating that, I ask students to tell me what 3x equals if x equals, 7, -3, 5/3, and so on, so they get the idea that values for the variable are substituted, and that 3x means multiplied by 3.

Division.  So far in previous lessons and discussions, whenever we’ve talked about fractions, we have mentioned that fractions are division.  So 5/2 is the same as 5 divided by 2; 2/3 is 2 divided by 3 and so on. I tell students that in algebra, we are going to represent division by a fractional form, rather than by using the symbols they have been using. The divide sign, ÷, is not used, particularly when using letters.  The expression x ÷ y is written as x/y. Similarly, x ÷ 3 is written x/3; 3 ÷ y as 3/y.

Examples for them to solve. I will write on the board various expressions with the values of the variables given for each problem, and ask them to find the value of each. These include expressions such as 5/x where x = 2.   5ab where a = -1, b = 2;  z + 2x where z = 5 and x = – 2, and so forth. I instruct them to leave improper fractions in that form; it is not necessary to convert them to mixed numbers.

Ending the Lesson. While the lesson may move fast, much of the information is new.  It is good to focus on these basics so they are familiar with combining like terms and being able to write expressions such as “three times some number” and “the sum of two different numbers” as a + b.  The next lesson will focus more on translating more complex English expressions into algebraic ones as well as the use of parentheses and order of operations.

Homework may include evaluation problems such as:

When y = 2, evaluate the expressions:

  1. y + 23 2. 6y            3.  8/y          4. 4 + y – 7

When m = 8 evaluate the expressions:

  1. 3m                       6. m/m                  7.  m x m x m

Traditional Math (10) Complex Fractions (7th Grade)

This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!

One thing that I almost never do is post the day’s “learning objectives” on the board. I find it sufficient to say to the class “Today we’re going to learn about…” and then say whatever it is we’re going to learn.  That seems to be enough.  There are occasions though when I will tell them what type of problem they will be solving at the end of the particular lesson.  I did this when teaching the topic of complex fractions to my class of seventh graders.

I announced that at the end of the lesson they would be able to do the following problem (which was a challenge problem that appeared in the JUMP Math teacher’s manual):

I was expecting to hear gasps and exclamations of “No way!” when a boy raised his hand and said “Oh, I know how to solve that.” He then narrated what needed to be done. He had certainly never seen this exact same problem before. He put together basic skills that he learned and saw how they fit together and solved a more complex problem—an example of knowledge transfer. Which is what this lesson is about, though most students will not be able to solve something like this straight off like this student did.

Warm-ups.  The warm-ups for this particular lesson should focus on what we have been doing, as well as some word problems:

1. (-2/7 + 5/14) ÷ 3/28   Answer:  1/14÷3/28 = 1/14 x 28/3 = 2/3

2. (2-3)/(2-7) Answer: -1/-5 = 1/5 

3.  -3/4 x 5 ¼  Answer:  -3/4 x 21/4 = -63/16 or -3 15/16

4.  How many 2/3 ounce servings are in a 5/6 ounce cup of yogurt?

          Answer: 5/6 ÷ 2/3 = 5/6 x 3/2 = 5/4 o 1 ¼ serving

Basic Lesson. Since we have just finished a lesson that covered fractional division, I will ask students to solve something like 2/3 ÷ 4/15 which they can do fairly readily.  They have learned in previous lessons that fractions are division. The fraction 6/2 is a division: 6 divided by 2. Similarly 2/3 is a division: 2 divided by 3. Therefore we can represent a fractional division problem as a “complex fraction”. The problem just given can be represented as:

Examples, Worked and Otherwise. We then practice rewriting complex fractions as ordinary fractional division problems, and solving them such as:

After a few of these, the problems can be more complicated:

And the solution to the first problem given (which my student solved without any lesson):

In general, my students enjoy complex fractions and look at them as puzzles. This is now something to add to the repertoire of problems to include on future warm-ups.