8.26.2011

Confession...I'm planning to use Khan Academy this year

It seems Khan Academy (henceforth referenced as KA) has a pretty bad rap in some math teacher circles.  I understand that the videos are somewhat lacking in the engagement factor, and motivating students with points and badges can seem somewhat elementary.  I also see that KA tends to focus more on a procedure/pattern than actual problem solving. 

All that said, I will be using KA this year in my resource classroom.  I have a group of students that are in my class for numeracy skill building/strategies instruction.  I'm supposed to be teaching them 25 mins/day and allowing them 25 mins/day to work on their homework or classwork.  I have students of all grade and ability levels in one resource class, so lesson planning becomes difficult.  Twenty-five minutes is not a long time when you think about it, seemingly less when you think about real problem solving tasks.

Enter KA.  Each student can be working on exercises related to what their individual math course is or will be addressing.  I can monitor their progress as a "coach" and assign them individual tasks.  My district uses Google Apps for Education, so all the students have email addresses that they use to log in.  I can, in theory, email them instructions for which exercises or videos I would like them to do during their 25 minute instruction period.  Two days in, some students are more engaged than others.  That is to be expected.  When assigned topics directly related to material they were working on in their math class, the students were more engaged.

I'm sure that things will change throughout the year as we continue to develop this strategies portion of the resource class, but for now, KA is a useful tool.  Just sayin'.

p.s. I (obviously) made it through week one at the new school.  Yay!

8.09.2011

Transitioning...

For those of you that don't know, I'm about to start teaching at a new school in a new state.  I will be still be teaching high school math to students with special needs, but those special needs are not necessarily deafness or blindness.  I am very excited to work with a math department as opposed to one other colleague (even though she was fabulous).  All in all, though, it's a lot of transition.

Here are some things I have learned in the process:
- I'm not as good at transitioning as I like to think I am
- I have a difficult time making any kinds of decisions when there are so many unanswered questions and uncertainties ahead
- It's really difficult to transfer certifications between states
- I'm not sure I entirely know what I'm getting into, but I'm still excited
- I am *not* a detail person

I went to a workshop hosted by the district I'll be working in.  It was great to collaborate with teachers from the different schools and create materials.  I should get back to working on my post-workshop assignment, but I hope to start up blogging again this year.

7.21.2010

Help!

Here's the deal:  I'm working on curriculum for my school and Algebra 2 is making my eyes cross.  I think the major problem is the state of Virginia is in a transition year between "old" Standards of Learning (SOLs), and "new" ones.  This year is supposed to be the year that we're still teaching and assessing the old SOLs, but we're supposed to teach the new ones, too.  Those of you that teach Algebra 2 already know that there's an enormous amount of information to cover in a short period of time.  To give you context, our school teaches it as a semester-long block course.  There's only so much a brain can handle in one day, though! 

Here's the first draft of my skills list and structure...I'm not sure what to do about the old vs. new SOLs (my skills list is based on the old SOLs because that is what will be assessed).

Note:  Gray items are not included in old or new SOLs but might be necessary for student understanding
          Blue items are being taken out of the SOLs starting next year
          Red items are new to the SOLs starting this year

Unit 1 Algebra 1 Review/Solving Equations


1 Solve multi-step equations and inequalities
2 Matrix +/-
3 Solve compound inequalities
4 Solve absolute value equations
5 Solve absolute value inequalities

Unit 2 Polynomial Review/Add Depth

6 Factor trinomial a = 1
7 Factor trinomial a > 1
8 Factor special cases (sum/diff of cubes, diff of squares, perfect square trinomials)
9 Factor out GCF first (factor completely)
10 Exponent rules
11 +/- polynomials
12 Multiply polynomials
13 Divide polynomials

Unit 3 Rational Expressions

14 Identify undefined values
15 Simplify rational expressions by factoring and canceling out common factors
16 Multiply and divide fractions
17 Multiply and divide rational expressions
18 Add and subtract fractions
19 Add and subtract rational expressions
20 Simplify complex fractions
21 Solve rational equations

Unit 4 Radicals, Radical Equations and Complex Numbers

22 Simplify numbers under radical
23 Simplify monomials under radical
24 Multiply and divide radicals
25 Add and subtract radicals
26 Nth roots to rational exponents and vice versa
27 Simplify expressions with nth roots and rational exponents
28 Solve radical equations
29 Simplify square roots with negative terms inside radical using i
30 Add and subtract complex numbers
31 Powers of i
32 Multiply complex numbers

Unit 5 Functions (intro)

33 Domain and range of relations (from ordered pairs, mapping, graph, table)
34 Identify relations that are functions and one-to-one
35 Given graph and a value k, find f(k)
36 Given graph, find zeros
37 Given graph and a value k, find where f(x)=k

Unit 6 Linear Functions


38 Slope from graph, equation, points
39 Graph from equation
40 Equation from graph
41 x- and y- intercepts
42 Determine whether lines are parallel, perpendicular, or neither from equation or graph
43 Write equations for parallel and perpendicular lines given line and point off the line
44 Graph linear inequalities

Unit 7 Systems


45 Solve systems of equations by graphing
46 Multiply Matrices using a graphing calculator
47 Inverse matrix method of systems
48 Systems of equations word problems
49 Graph systems of linear inequalities
50 Linear programming max/min problems

Unit 8 Functions (reprise)

51 Function math (addition, subtraction, multiplication, division)
52 Function composition, find a value i.e. f(g(3))
53 Function composition, find the function i.e. f(g(x))
54 Find an inverse function by switching variables

Unit 9 Quadratics

55 Graph from vertex form, identify max/min and zeros
56 Solve by factoring
57 Solve by Quadratic Formula (including complex solutions)
58 Determine roots using the discriminant
59 Write equation for quadratic given roots
60 Quadratic systems
61 Polynomials: relating x-intercept, zeroes and factors
62 End behavior for polynomials

Unit 10 Exponential/Logarithmic functions

63 Exponential growth or decay from function
64 Sketch base graph of exponential/log functions
65 Exponential to log and vice versa
66 Data analysis/curve of best fit for linear, quadratic, exponential and log

Unit 11 Transformations and Parent Functions

67 Graph absolute value functions
68 Horizontal and vertical translations of linear, quadratic, cubic, abs value, exponential and log
69 Reflections and stretching of linear, quadratic, cubic, abs value, exponential and log
70 Combinations of transformations on parent functions
71 Identify parent graphs of parent functions
72 Identify equations of parent functions

Unit 12 Conics

73 Identify a conic from graph
74 Identify a conic from equation

Unit 13 Variations

75 Write equation for direct, inverse and joint variation problems
76 Find the constant of variation

Unit 14 Sequences/Series

77 Write n terms of an arithmetic sequence
78 Find the sum of a finite arithmetic series
79 Write n terms of geometric sequence
80 Find sum of geometric series
81 Use formulas to find nth term
82 Identify sequence/series as arithmetic, geometric or neither

Unit 15  Statistics

83 Determine probabilities associated with areas under the normal crve
84  Compute permutations and combinations

 
If you made it this far, here's my call for help:  Anyone have advice/suggestions for how to make this work and/or a better way to organize the information into cohesive units that seem to occur in a somewhat logical order?  There is and will continue to be an emphasis on function families and transformations (as there should be).  I find it difficult to express on paper how each function category needs to be a resting place, but they are all connected in the ways that transformations apply.  Any ideas?
 
...oh...and I'm going to be teaching one section of deaf students and one section of blind students...in case that makes a difference

**edit:  I've added links to the old and new Virgina SOLs for Algebra 2 if anyone's interested**

5.21.2010

The Birth of an Assessment

Sam recently blogged looking for feedback on an Algebra 2 assessment he gave, but mainly to start a conversation about assessment creation, etc.  My classroom assessments have changed drastically in my short 2 years of experience.

Here's a chronology of my growth:

1st year teaching (Algebra 1 - full year course)
  • Classroom instruction followed the sequence (and pacing to some extent) of the textbook we were using (McDougall Littel Algebra 1).
  • Assessments were largely based on Chapter Tests from the end of whatever chapter we were in, re-typed/formatted but using the same problems with little thought to balance what was actually being tested.
  • Points were assigned to each problem to award partial credit for being on the right track (this often ended up meaning the more difficult problems were worth more points than the basic problems - more steps = more points)
  • Here's an example of one such test: Chapter 9 Test.  Point distribution is as follows (side note: what was I thinking with these point distributions...they make no sense!)
    • #1 - #4 :  38 points total (one for coefficient and one for variable in each term of each problem)
    • #5 - #6:  6 points each (coefficient part, variable part for each term)
    • #7 - #8:  4 points each (2 for multiplying correctly, 2 for combining like terms correctly)
    • #9 - #10:  10 points total (coefficient part, variable part for each term)
    • #11 - #12:  9 points total (1 pt each - GCF number, x, y*for #12*, each term left inside parentheses)
    • #13 - #16:  4 points each
    • #17 - 20:  6 points each (4 for factoring, 2 for solving) *#19 was eliminated as unfactorable...oops*
    • Total out of 100 points (this was a rare occurrance...my tests rarely end up being "nice" numbers of points)
Looking back, getting ready for year two, I realized that the tests I gave in year one were nothing more than a random assortment of problems that may or may not have given me an idea into what level of understanding my students had.  Mostly, it was a hoop they (and I) had to jump through at the end of a chapter that didn't tell either one of us anything relevant.  If a student failed a test, I would work with them 1:1 to try and remediate some areas, but it was haphazard at best.  If the whole class failed, then I would plan a retest.

2nd year of teaching (I'll stick to year-long Algebra 1 course for discussion purposes, despite having other classes as well)

- Fall Semester
  • Classroom instruction divided into more logical "units" instead of staying strict to the chapters in the text.
    • Unit 1: Number Sense and Properties
    • Unit 2: Variables, Exponents and Substitution
    • Unit 3: Solving Equations and Inequalities
    • Unit 4: Proportional Reasoning
    • Unit 5:  Graphing
    • Unit 6: Writing equations
    • Unit 7: Systems
    • Unit 8: Polynomials
    • Unit 9: Factoring
    • Unit 10: Quadratics
    • Unit 11: Statistics
  • Assessments were created with more thought given to the types of questions and what they were really asking/showing to give balance
  • Points were assigned for each problem for partial credit opportunities, but with consistency and balance in mind. (i.e., being careful not to weigh a really difficult problem too heavily or allow one specific skill to dominate the point distribution)
  • Here's an example of a unit test:  Unit 2 Exam. Point distribution is as follows:
    • Simplify: 20 points total
      • first row - 1 pt each (basic exponent rules)
      • other problems in this section - 3 pts each
    • Evaluate: 2 pts each (1 for substituting, 1 for simplifying)
    • Let a = 3...: 3 pts each (2 for substituting, 1 for simplifying)
    • Order of operations: 3 pts each
    • Total out of 62 points

- Spring Semester

At the end of 1st semester, my colleague and I started talking much more about assessment.  A lot had changed compared to my first year, but we still weren't satisfied with the information we were getting from students' test grades, nor were the students doing anything to study or improve their skills.  We sat down during our "Snowpocalypse Week" and hashed out a standards based grading system.

Based on the needs of our students, we decided that assessments now would have two forms:
  • Individual skills assessments
    • focus is on progress towards mastery of one learning target
    • points are assigned with a 1 - 5 rating for each skill.
      • 5 = A, mastery level: true mastery of high school level content
      • 4 = B, mastery of the basic level content
      • 3 = C, general understanding of basic level content, some mistakes
      • 2 = D, several mistakes or holes in understanding
      • 1 = F, many mistakes or solutions off point
    • Here's an example of a skills assessment used this term. (basic level, mastery level) The page has 4 skills, but each receive an individual grade
  • "Retention Tests"
    • focus is on whether or not a student is able to recognize solution strategies when there is more than one problem type being addressed 
    • is the student is retaining the information from previous skills targeted?
    • points are assigned in a similar manner to the fall semester unit tests
  • Instruction is still in "units" and retention tests happen at the end of units. 
  • Skills assessments happen on a more ongoing basis: teach, practice, skills assessment, reteach if necessary, re-assess as needed, etc.
In general, I like having at least one question on a retention test and/or on the mastery level of a skills assessment that isn't exactly like any other problem they have seen.  These types of questions help students synthesize what they have learned and help me see if/how they can apply the strategies in new contexts instead of just regurgitating a procedure.

This year has certainly been one of growth! Some fruit of our labor in the field of standards based grading? Nine of twelve students in Algebra 1 passed the state End of Course Standards of Learning assessment on the first try.  The other three qualified for an expedited retake.  Two of them passed.  This means we have a 90% pass rate for first time testers in Algebra 1 this year (9 out of 10...2 students that passed this year were re-taking the class)! Last year it was a mere 37.5% (3 out of 8).  While I am aware that passing the state test is not a true picture of whether or not the students can *do* Algebra, I also know that they were more prepared for the test due to targeted remediation of their weak skills as shown on the skills assessments.  That's an achievement.

5.01.2010

ASL/English Vocabulary in the Math Classroom

My last semester in college, while I was student teaching, I had a class that emphasized different key topics in the field of Deaf Education.  One such topic was vocabulary development.  We all already knew that students who are deaf/hard of hearing have a lower vocabulary than their same-age hearing peers for a variety of reasons not least of which being their limited access to "incidental learning" that comes from listening to other people's conversations/tv/radio, etc.  In our class, we talked about ways to introduce new vocabulary in order to give students a more connected understanding of the new word in its five distinct forms.
  1. Picture
  2. Description/definition
  3. ASL sign (if applicable)
  4. English word (in print)
  5. Fingerspelling of English word
I try to be conscious of this as I teach.  It's very difficult sometimes, and many of the math terms to not have standard ASL signs, so it is more difficult for the students to attach meaning and use the new term through fingerspelling alone.

In Calc this week, I had my student doing practice AP Free Response questions.  One day, after completing a no-calculator free response question requiring justification of responses, I read the justification and was reminded of vocabulary difficulties.  The mathematical justification was great, but instead of saying the normal line is perpendicular to the tangent, therefore the slopes are opposite reciprocals, justification was that the slopes are "negatived and flipped."  In ASL, it would be an appropriate explanation, because the sign for flip and the sign for reciprocal are the same. This is the class that I have been most conscious about vocabulary! We have had English lessons in the middle of calc class in order to recognize the different forms of words that have the same sign.  i.e. differentiate (v.), derivative (n.), differentiable (adj.), derive (v.), etc.  The majority of the calc vocabulary (inflection point, slope, tangent and many others) does not have ASL signs, so we do a lot of fingerspelling with the support of written English on the board.  I am concerned that this might not be enough.

Any  thoughts on how to make vocabulary a more essential part of the curriculum or to get students actively using appropriate terms in writing?