Reflective Journal Entry 1
November 17, 2003

Last week in College Algebra we were working with conic sections.  We would spend a day working with each one and what the equations look like and then we would spend the 2nd day doing application problems in pairs or small groups.  I would categorize the applications as doing the math because even though they knew what the equation should look like, they had to do different things to come up with it.  They worked in small groups and helped each other out a lot.  They would grab graph paper as needed and refer to formulas if needed.  I was very pleased with how well and hard they worked.  They helped each other out and sought out my assistance very rarely.  By the end of the unit, they did not need the formulas at all, they had worked enough problems to figure out the symmetries and similarities and differences in the different types of conic sections.  I felt that the application days stayed at doing math.  Over break I will grade the tests and hopefully I will still be singing their praises.

Reflective Journal Entry 2
November 25, 2003

In Algebra 2 we are learning how to factor.  In the first worksheet I divided the problems up into the different types as a review of factoring that they did in Algebra 1.  That worksheet would fall into procedures without connections.  The problems were placed according to the type and they just basically had to follow the algorithm.  There were however a large number of students that could not seem to grasp the notion that all the difference of squares problems were in one column and therefore it was much harder for them.  Those students were then at using procedures with connections.  Today, the day before break, we played factoring bingo.  I feel that today they were using procedures with connections.  The factoring types were all mixed together.  So even though there was a procedure they first had to figure out what procedure to use.  The students worked hard at it because candy was at stake.  It was a very productive day before break!!

Reflective Journal Entry 3
December 8, 2003

In Honors Algebra 2 I used an activity that I found in the Mathematics Teacher.  It was an activity where the students had to make a number line and mark off irrational numbers along with some rational numbers but they could not use a ruler or a calculator.  First they had to construct a square and a 30-60 right triangle.  The side of the square was the same length as the short leg of the 30-60 right triangle. Using the side of the square as on unit, they marked off a number line from -3 to 6 on a piece of adding machine tape.  Then I gave them a list of irrational numbers to mark off just using the square and the triangle, for example:  square root of two, one plus square root of three, two square roots of two, three plus square root of three, and square root of three divided by two.  It was a great exercise because there was a lot of review from geometry and also they learned something new and hopefully realized that irrational numbers are actually numbers on a number line not just unfortunate answers.  I felt that it fell into the procedures with connections because there was a suggested pathway to follow but they had to rely on conceptual ideas in order to actually find the correct marks. They had to figure out to make folds on the sides of the shapes in order to get for example square root of three divided by two.  They did a great job and had some good discussions with each other.

Reflective Journal Entry 4
January 15, 2004

Today my Algebra 2 students worked on a graphing calculator lab.  They had to analyze quadratic equations and determine how the a, b, and c values affect the graph.  First they analyzed equations in standard form (y=ax^2+bx+c) and then they switched to vertex form (y=a(x-b)^2+c).  I didn't know if that would be confusing switching between the 2 forms, but they handled it well.  Everyone had a graphing calculator and they would put 4 different graphs in the calculator at a time.  In that set of 4 equations they had to keep for example a and b constant and just change the c and then tell how the graphs were different and how they were the same.  I would have to say that here they were working at the procedures with connections.  They were following the procedure of putting the equations into the calculators in order to develop a level of understanding of the affect the different changes had on the graphs.  I was very pleased with what they came away with during that activity.
Note:  This is the following Wednesday and a majority of the students remembered what they had discovered last week.  Yeah!!

Reflective Journal Entry 5
January 22, 2004

This week feels like I didn't get much teaching in.  We didn't have school on Monday and I was out Tuesday observing in elementary schools in the St. Vrain district.  Our Math Study Group is researching textbooks to buy for next year, so we are visiting schools that are currently using textbooks that we are interested in.  Today we did a Lesson Study and I was out of my own classes again this afternoon.  Like I said, I haven't done much actual teaching this week.  Anyway, today my College Algebra students were working on writing a series in summation notation.  The series that end up being arithmetic or geometric are easy for them to do, they just have to use the formulas.  In those cases they would be working at procedures with connections, assuming that they recognize that the series is arithmetic or geometric.  They are really complaining about the series that are neither arithmetic or geometric.  In those instances they have to look for patterns and guess and check.  They are at the doing math level because they do not have a predictable, well-rehearsed approach or pathway.  Also, you know that they are doing math when they get all upset about it.  That is where they were at today.

Reflective Journal Entry 6
January 26, 2004

Today in Honors Algebra, we had a blast.  We spent all hour on only 4 problems.  It was actually meant to be a review of a concept from the chapter but it ended up being a whole new lesson.  We are working with rational expressions and particularly graphing them.  I decided yesterday as I was planning the review questions, to show them the graphs and have them come up with the equtions.  I had graphed four different rational expressions on a piece of graph paper and asked for these things:  a)vertical asymptotes, b)horizontal asymptotes, c)x-intercepts, d)y-intercepts, and e)possible equation.  We worked the first one together and then I let them try it on their own.  I let them struggle with them.  I would give them hints along the way but they were all doing mathematics.  The whole class was very engaged and tried to help each other out.  It was really cool and I felt that they all left with a better understanding of the graphs and how they relate to the equations and vise versa.  Score!!

Reflective Journal Entry 7
March 3, 2004

The week of February 12th I spaced off the journal so I will make it up today.  In jounal entry #10 I was talking about an activity for composition of functions and I noted in there that I would do the exercise that Dr. Farmer shared with us in a class the next day.  Well that day was yesterday and in my Honors Algebra 2 class I did try the activity where you have three groups and ask the first group to come up with a list of input values that they give to the next group.  That group comes up with a function and applies the inputs to the function.  They give those outputs to the third group that comes up with a different function and applies the 2nd group's outputs as their inputs.  I had a student up the original inputs and the final outputs on the board as a table of values and asked them to come up with the rule.  The first function was y equals the square root of x and the second function was y=2x^2+6, so the composition ended up being linear.  Most of them had a cow and insisted that they would not be able to do it.  One young lady after about 2 minutes had a rule.  The others felt stupid when they found out that it was linear.  It did not even dawn on the kids to graph the points.  I felt very inadequate when I saw how troubled they were with a fairly simple problem.  Next I put up a table of values of an absolute value function and had them come up with a rule.  About half of the figured that one out.  I scraped the whole idea of doing compositions and just worked on functions.  I will be coming back to that activity again with that group.  I will really have to get my courage up to do it with the Algebra 2 kids, but it is something that has to be done.  I do think that as they worked on writing the functions they were doing math.

Reflective Journal Entry 8
February 19, 2004

In Trigonometry we were working on the unit circle the last couple of days.  I told the students that the unit circle will not be able to be used on quizzes or tests and that the unit circle that we made is just a tool of reference to help them understand how to get the exact values of trig functions of the special angles.  Today they were finding the trig functions without the unit circle in front of them.  Before I grabbed the frameworks I was thinking that this activity would fall into procedures without connnections.  I thought that the activity was pretty mindless, even though some of the students really had to work at it.  As I sit looking at the frameworks I feel that it satisfies more of the bullets under procedures with connections.  I do think that you use a procedure to help develop deeper levels of understanding of the concepts.  The procedure canot necessarily be followed mindlessly, you have to keep track of what quadrant you are in so that you get the signs correct and you need to make sure that you get the correct reference angle

Reflective Journal Entry 9
February 23, 2004

Today in Algebra 2 they were solving equations using quadratic techniques.  I had a chart that they were filling out.  I would start with a quadratic equation that would factor and they would solve the equation and then in the next box they would solve an equation in quadratic form that factored the way the quadratic did.  I wanted them to see the similarities between the two.  They did quite well.  They figured out the pattern of equations in quadratic form.  They noticed that you had to square the term but that translated to doubling the exponent.  I was tickled that they came to that conclusion on their own.  They could quickly tell if an equation was in quadratic form and then most of them could also factor it correctly.  The most trouble came at the next step.  How do you solve an equation such as x^(2/3)=16?  We talked about changing it to radical form and then doing the reverse operation to solve.  The challenging part was remembering to add a plus or minus sign if you were taking the square root and not adding it if you were taking the cube root.  I was tickled that they could get it factored and could get at least most of the way through it.  It is amazing how often the students take a problem that should be a lower level task and make it a higher-level task because they don't have a very good background of basic algebra skills.  It is interesting that as we look at making the schedule for next year, we have second year teachers that are anxious to teach an Algebra 2 class because they think the students will be so much better.  I think that they will be shocked to find that teaching an Algebra 2 class when the Algebra 1 teacher hasn't done a good job is not all that glamorous.  I really don't think that I will tell them that, that is something that they will have to learn the hard way.

Reflective Journal Entry 10
March 1, 2004

I am about going crazy in Algebra 2 with the chapter on polynomials.  It is very hard for the kids because I do not show a lot of application because there are not too many good ones.  I think that it can be very hard for them trying to keep it all straight.  I did find a worksheet on composition of functions that I found to be kind of cool.  They started off with these five functions.
Lin(x)=ax+b, enter into Y1
Quad(x)=x^2, enter into Y2
Sqrt(x)=square root of x, enter into Y3
Abs(x)=absolute value of x, enter into Y4
Recip(x)=1/x, enter into Y5
Then they would be given a function that was a composition of two of those functions, for example f(x)=9x^2-11.  For this particular function, it is a composition of lin(quad(x)).  After they had an idea of what it was they then had to check it on the calculator by putting the actual function into Y6 and then the composition Y1(Y2) into Y7 and then checking a table of values to make sure it was the same.  Because this particular function used the linear function, they would haave to enter 9x-11 in Y1.  I think it was a good warm up for compositions.  The task itself seemed to be pretty easy for the most part.  The task analysis would be a procedures with connections.  In the calculator they witnessd two representations of the same composition.  The procedure that they were following was a general procedure that helped them with the underlying concept.  Tomorrow in class I will be doing the exercise that Dr. Farmer did with us in class using composition of functions.  I am looking forward to that.  I will be doing it with my honors class first to see how they handle it.

Reflective Journal Entry 11
March 17, 2004

Last week slipped by me so I will do two entries today.  In Honors Algebra 2 we are working on conic sections.  Actually we just started and are using the distance and midpoint formulas.  We are also working with points in 3-D space.  I got out the isometric dot paper and we were graphing in 3-D because most said that they had not done that.  Most of the problems required them to use appropriate formulas to come up with the answer.  One problem asked them to name the other verticies of a box given two points that were in opposite corners.  They were using the ordered triplets (4, 1, -1) and (2, 3, 5).  I had no idea that it would be so hard for them.  Okay, one kid had a program written by the end of the hour but the others really struggled with it.  They took it home because we ran out of time.  I am sure that a few didn't look at it again that night but most of them did and just a handful came up with the other verticies.  The were working at the doing math level because most of them did not have a clue where to start.  They had isometric dot paper and were trying to graph but could not make sense of it.  They were very good at sticking with it.  One kid came up with the points and then from that came up with a way of doing the problem without graphing.  It is interesting seeing the different ways that different kids will approach a problem.  I had each one of them present what they did if it was different from the last.  That activity went much deeper than I thought it would.

Reflective Journal Entry 12
March 17, 2004

In Trig class we have been working on graphing the trig functions.  They know how to find the amplitute, vertical shift, phase shift, and period for graphs of the trig functions.  Today we started on some application problems.  I have a packet of dat and they will have to come up with an equation to fit it.  The data varies from average monthly temperatures, to height on a ferris wheel, to distance away from a motion detector of a pendulum.  They will have to graph the points and then come up with the equation of a sine curve.  I believe that they will be working at the procedures with connections level for the most part.  The students will need to engage with the conceptual ideas that underlie the procedures in order to complete the task.  They have only worked from the equation and then identified the changes.  Starting with a set of points and working backwards required some thought.  Most were overwhelmed and I had to break it down for them.  I told them to just figure out the amplitude first and then move to the vertical shift and so on.  That helped out a lot.  For the first problem many of the students experienced doing math.  They had to access relevant knowledge and make appropriate use of it.  I do think that the more they do the easier the task becomes and the level drops.  We are not at that point yet, so they will likely struggle with it tonight.
 

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