Summer 2012 Goals

I found out I should be teaching the same subjects next year- hoorah for no new preps!
Also, my instructional coach has written my End of Course exams for Algebra I, Geometry, and Algebra II based on Common Core standards and including performance events and extended response. This will be the backbone of my curriculum and my guide for everything I want to do. Here are my goals to accomplish this summer:

• Design math portfolio (I'm actually doing this for my master's so hopefully I will be done with that in early May)
• Tweak and tighten my pacing guides from last year
• Use the pacing guides and backwards design strategy to create unit plans with essential questions and key concepts list
• Create performance events and a rubric for each unit
• Use the above to finally implement standards based grading
• Create questions/system for Lee Jenkins formative assessment idea
• Fix sucky lessons

I know and accept the fact that I will not fully accomplish all of these things. But my list is the shortest summer list I've had yet which is a good sign. We get out of school the 21st, about 2 weeks earlier than usual. The rest of that week is a teacher work week so I hope to tackle as much of this as possible in that week before I lost my motivation.

I'm taking two grad school courses this summer (I'll be two-thirds done at the end of summer!!) so I'm not going to pretend that I will do much classroom stuff after June 11th.  Nevertheless, I'm feeling excited, underwhelmed, and counting down the days.

43!

SBG Exemplar and Rubric

I hate the new Blogger interface. It deleted my post that was supposed to be a draft and then published it so that only a title showed up. Boo. Luckily I can still revert to the old Blogger interface. If you clicked through to this before and saw nothing, my apologies.

After my SBG brainstorm post, I really started to zero in on what kind of rubric I want to create. The main thing I want my rubric to do is to easily show anyone who looks at it (parent, student, teacher, administrator) exactly what the student has done right and exactly what the student has done wrong. I want my rubric to give clear feedback and possibly even cues or questions that instruct students on what to do next.

Is this even possible? I think the thing that has been holding me back is that I thought I could create a rubric that would work for everything (ala a writing rubric, a team work rubric, etc). But that is just not true. I'm looking for different skills for each concept I teach and so the rubric must be adjusted accordingly. Another hint that I must do backwards design: I have to clearly outline the outcomes I want students to achieve (a key concepts list), the method I will use to determine if they learned it (the performance event), and the method of communicating to students if they were successful (the rubric).

Another thought that has held me back is thinking that I should be able to look at a student's grade and immediately know where they went wrong without looking at the rubric. I know, I know that makes no sense. But it kinda does. If you create a rubric like I've seen elsewhere, such as

• 4- Performs skill with no errors. 
• 3- Perform skill with minimal errors
• 2- Performs skill with major errors
• 1- Cannot perform skill

then technically, you could know how much a student messed up, but not at what exact point. So my new idea is actually better than the idea I was trying to live up to! Twisted.

I really liked Lizzy's comment about giving students one point for different steps in the process like the AP world does on a regular basis, Kristin's comment about giving students a basic, proficient, and advanced problem, and Adam's comment about having students use rubrics to make test corrections.

I'm going to combine these ideas into a giant megamind idea!

My exemplar will continue to be on system of inequalities and will now be accompanied with pictures.

Here we go:

#1 Basic

#1 is the most basic problem. Students would be graded as follows:

• Graphing inequality 1 with correct slope and y-intercept (2 points)
• Graphing inequality 2 with correct slope and y-intercept (2 points)
• Determining if inequality 1 is solid or dotted (1 point)
• Determining if inequality 2 is solid or dotted (1 point)
• Plugging in test point for inequality 1 (1 point)
• Plugging in test point for inequality 2 (1 point)
• Shading in the correct direction for inequality 1 (1 point)
• Shading in the correct direction for inequality 2 (1 point)
• Stating an ordered pair from the solution set (1 point)

These are the most basic skills to solving systems of inequalities. #1 is worth a total of 11 points.

Moving on to problem #2:

#2 Proficient

The only difference in this proficient problem is that now the inequalities are in standard form and must be put into slope-intercept form. Grading as follows:

• Solving inequality 1 with correct slope and y-intercept (2 points)
• Solving inequality 2 with correct slope and y-intercept (2 points) 

#2 is worth a total of 4 points. Students will only be graded on the new part of the problem because they have already completed the basic skills on #1. Students won't know this and won't know that I'm only grading part of it. This prevents students from doing the bare minimum.

On to #3:

#3 Advanced

The new skill in the advanced problem is that the student must write the system of inequalities from a word problem. Grading:

• Writing inequality 1 with correct slope, y-intercept, and inequality sign (3 points)
• Writing inequality 2 with correct slope, y-intercept, and inequality sign (3 points) 

Students will then have to solve (like they already did on #2), then graph (like they already did on #1.) Problem #3 is worth a total of 6 points.

The entire performance event would be worth 21 points.

Students would receive a rubric at the beginning of the unit to monitor their progress along the way and to give feedback. They would check off skills as they learn them and write the date. I could use this rubric to grade quizzes along the way, using the bottom section for specific feedback. Then I would use the same rubric to grade their performance event at the end of the unit. 

Rubric:

Rubric

Does this seem clear and easy to understand?

Do the grade ranges seem fair? For example, a proficient student scoring 12 out of 21 points would receive a B on my rubric but normally a 12 out of 21 would be a 57%. That feels crazy. Am I totally missing the mark here?

I actually did grade this way, I just didn't give students a rubric. The assessment really was worth 21 points and I really did just give percentage grades like the aforementioned. The difference is that I would now be clearly communicating what I expect students to be able to do and clearly communicating how I am assessing those abilities. The grade is the only thing that is throwing me off. Am I out in left field here?

SBG Brainstorm

Next year is my last year of having an instructional coach so I really want to get all of my big ideas out there on the table by next year so that I have some kind of help and guidance when things go wrong.

I am always thinking about SBG and how I can make it work. I re-read my last post on SBG, which was almost a year ago. I think that might now be possible. My math teacher counterpart is leaving after this year and I'm thinking the administration might consider hiring the current student teacher. If so, it will be their first year and so far they seem really...'moldable'. And I think any first year teacher would love to have another teacher give them resources, ideas, and collaboration. If this is true, we might be on the same page with SBG and so on!

Anyway, my brainstorm is this. My number one struggle with SBG has been not assessing everything I teach. In my mind, I feel like if I'm teaching it, why wouldn't I assess it? Now I'm thinking that I just need to cluster. Concepts build on top of each other toward an ultimate goal. If I only assess that ultimate goal, then it requires knowing the concepts that build up to that goal without assessing every little thing individually. I would call this ultimate goal a performance event and use a rubric idea to grade it. A perfect score would be getting every skill correct. Otherwise, each skill would move you one step higher on the rubric.

Here's my example based on a mini-unit of solving systems of linear inequalities. The ultimate goal of the unit would be to graph a system of inequalities from a word problem.

Skill 1: Write a system of inequalities from a word problem.
Skill 2: Solve each inequality for slope-intercept form.
Skill 3: Graph both lines of the system (dotted or solid).
Skill 4: Plug in a test point and shade.
Skill 5. Choose an ordered pair from the solution set.

To earn a 5 for the ultimate goal, they would have to correctly complete skills 1-5. If they could only do 1 and 2, their score would be a 2. By comparing the rubric and the students grade, it should still be easy to remediate.

Here's where my idea breaks down. What if they complete every skill but they screwed up on skill 2? If they complete the subsequent skills based on their answer for skill 2, they may have done the process correctly but still ended up with a wrong answer. Would they then get a 2 because that is the level where they screwed up? Or would they get something like a 4.5 because they did the process correctly? Translating to the gradebook, if each skills was worth 20%, would the student get a 40% (2) or a 90%(4.5)? That's a huge difference.

Another option would be to count each skill as one point (20%) so that they get credit for every skill they performed correctly but then their score would tell me nothing about where they messed up. :(

Other than that, I am liking this idea. The individual skills will be assessed throughout the unit through class work and homework quizzes but the unit test would now be replaced with this performance event idea.

I also want to try math portfolios next year. I'm thinking (hoping) that this idea would naturally tie into the portfolio. After completing the performance event, the student would then write a structured reflection/explanation of the process. Throughout the year, these performance events and reflections would be collected as a demonstration of mastery.

One last thing we are working on are End-of-Course tests. This is not a state mandate but a district suggestion. We want it. We have currently finished our Algebra I EOC minus a few edits. We will give this four times a year to show growth over time. This will also help students be prepared for long, standardized tests as opposed to the short performance event quizzes covering only one concept.

Please tell me where you see potential downfalls and advise me on how to grade!

I want to make this workkkkk.

Ch 5 Literacy Strategies for Improving Mathematics Instruction

I do these more for myself than anyone else, but here, I am quoting the most useful parts of this ASCD book (click links to read online for free). Basically, I'm editing out the boring. You're welcome.

I'm also just doing a couple of chapters at a time because it's kind of dry and I never know how much time I will have to read. So consider it a series if you like.

Literacy Strategies for Improving Mathematics Instruction

Joan M. Kenney

Ch 5: Discourse in the Mathematics Classroom 

I really really liked this chapter so I'm pretty much typing the whole thing. But oh well!

For the purposes of this chapter, I am defining discourse as the genuine sharing of ideas among participants in a mathematics lesson, including both talking and active listening.


What Discourse Looks Like

• Traditional. Prompts from the teacher generally lead to a preplanned answer. I do this.
• Probing. Stem from the teacher's desire to hear about students' thinking, rather than from a need to move students along a planned route. I do this too.
• Discourse-Rich. "Students must...learn to question and probe one another's thinking in to clarify underdeveloped ideas. Not this. 

When the classroom climate fosters genuine student discourse, students react to their classmates; ideas, asking questions and checking for understanding. Conversations continue without the need for teacher participation or supervision.


Creating Discourse-Friendly Classrooms

• Arrange furniture so that students can easily turn to see each other. They must be able to speak and listen to classmates. How can I do this with 24 desks?
• Encourage students to direct questions and explanations to the class, rather than to the teacher. I usually redirect the question back to the class but have not actually told to ask the class instead of me.
• When recording ideas on chalkboard or chart paper, use the students' words as much as possible. This is a matter of respecting their ideas.
• Try not to repeat or paraphrase everything students say. Paraphrasing can give the impression that the student is being corrected and may indicate to others that they don't need to listen unless the teacher speaks.
• Remind students that conversation is a two-way operation requiring both talking and listening. 
• Stand in a variety of spots. As students turn to look at you, their views of the classroom and their positions relative to classmates will shift.
• Give students time to think. Wait time or brief writing moments help students to solidify ideas and formulate good questions.
• Arrange lessons so that students have a product to share as they explain their thinking. They might illustrate ideas on chart paper or overhead transparencies or demonstrate by using manipulative materials.

 Teachers must understand mathematics at a deep level in order to follow young people's unconventional reasoning. This is exactly what I am afraid of!

Here are several ways to let student ideas take the lead in class:

• Involve students in engaging and challenging problems.
• Ask open questions to stimulate student thinking. (Examples: "What does this make you wonder about?" "Are there patterns?" "Is this logical?" "Can we estimate a solution?")
• Listen carefully to student responses.
• Train students to listen to their classmates; observations by asking questions that engage.
• Honor diverse ideas, methods, and example from varied sources. 
• Honor ideas even if they're incorrect. Do not quickly agree or disagree. Students will come to realize that you are giving them time to think and to justify. Often, as students explain erroneous thinking, they uncover their own errors or classmates step in to clarify or correct them.
• Encourage mathematical arguments between students.
• Remember, confusion is okay. Some of the best learning happens when we sort out what it is that has pushed us a bit out of balance. Be sure students know you are deliberately letting them be confused and that this is based upon your knowledge of how people learn, as this tactic may not match what previous teachers have done. And they will think you are not doing your job!
• Take time to let students share different problem-solving methods. Even when a correct solution has been shown, ask if there are other ways to do the problem. This helps to deepen understanding and makes students more willing to work with their own strategies, rather than thinking there is only one correct method.
• Capture teachable moments. Tangents are good.
• Decide how much leadership your students need. Let students' ideas lead but this doesn't mean that the class just moves without teacher direction.

Tips for Facilitating Classroom Discussion

• Focus students attention on a problem, puzzle, figure, process, question, or set of numbers. Stimulate discussion by asking the following types of questions:
• What do you notice?
• Do you see any patterns?
• What is similar?
• What is different?
• How do you think this works?
• Why does this work/look this way/give this result?
• What questions do you have?
• What can we do with this information?
• What do you want to know?
• Rather than rephrasing their responses, ask, "How do you want to say this?"
• When observations or questions are brought up by one student, ask, "What do the rest of you think about this idea? Does it make sense?" Encourage them to consider other examples that would show that the observation is or is not always true.
• Motivate students to search for patterns, delve deeper, and generalize.
• If students are making mistakes or doing something awkwardly, ask them "Is there an easier or more efficient way?" or "In what other ways could this be done?" rather than telling them how to do it.
• If students have difficulty thinking about a concept, suggest examples to consider or play devil's advocate. Ask "What if?" questions. I love playing devil's advocate!
• Counter questions  with questions instead  of explanations. (Students hate this!) People tend to blank out when one person asks a question and the teacher immediately gives an explanation.
• Even when a solution is successful, take time to ask whether anyone did the problem a different way or discarded an idea. Help students to build confidence in their own ideas , knowledge, and insights by showing that problems can be solved in a variety of ways.

Discourse and Problem Solving

As students explain their thinking, others can see connections and the usefulness of different methods.

Summarizing and labeling the strategies make them memorable, as does naming them in honor of the students who came up with them (e.g., Mary's Method, Theo's Theory, Pedro's Plan).


Discourse and Vocabulary

During lessons in which students first encounter a new concept, teachers should encourage them to describe ideas in their own words before introducing the specialized terms.


Summary

As we change our teaching, it is important to realize that we expect students to change with us; they also have new responsibilities.

Students may well be uncomfortable shifting from passive observers to active learners. How can we help them learn these new skills?

--
Resource:

There are special coupons available for books on this topic.

Ch 4 Literacy Strategies for Improving Mathematics Instruction

I do these more for myself than anyone else, but here, I am quoting the most useful parts of this ASCD book (click links to read online for free). Basically, I'm editing out the boring. You're welcome.

I'm also just doing a couple of chapters at a time because it's kind of dry and I never know how much time I will have to read. So consider it a series if you like.

Literacy Strategies for Improving Mathematics Instruction

Joan M. Kenney

Chapter 4. Graphic Representations in the Mathematics Classroom

The following aspects of mathematical language are particularly confusing to students:

• Technical symbols such as ∑, ≤ , or Δ. These signs, also known as logograms, stand for whole words but have no sound-symbol relationship for students to decode.
• Technical vocabulary- words such as rhombus, hypotenuse, and integer, which are rarely used in everyday conversation.
• The assignment of special definitions to familiar words such as similar and prime.
• Subtle morphology (one hundred, hundreds-place, hundredths) and the use of "little words" (prepositions, pronouns, articles, and conjunctions) in a technical syntax so precise that meaning is often obscured rather than clarified.  

Scenario #1: Measuring Cups Activity

*A teacher models drawing a one cup and one-fourth cup four times to quadruple a recipe. When asked how many cups are needed, students responded 'eight' because there were literally eight cups in the drawing. They counted the one-fourth cup the same as the one cup.

• In what other ways might the students have attempted to model the problem if the teacher hadn't offered the initial suggestion? Let students try first, then offer suggestions?
• How might using actual three-dimensional models (i.e., a set of nested measuring cups) before using two-dimensional representations alter the transfer of learning? Always teach from concrete to abstract? 
   

Scenario #2: A Round Pizza in a Square Whole

*When solving a problem dealing with rectangular pieces of pizza, William drew the problem with a circle pizza. To him, fractional parts had to be round.

• How can the teacher help [William] move beyond a single conceptual image and experiment with new metaphors and visual models?
• When are models useful, and when do they get in the way of new learning?
  
  

Scenario #3: An Uphill Struggle (Slow on a Slippery Slope)

*When discussing slope from points on a graph and connecting points to create intervals, a student took the graph literally, thinking the the intervals were literally hills and valleys.

• For how long, and to what depth, should a teacher continue to probe in order to get to the logical and intuitive root of confusion?

Scenario #4: Breaking Even

*Students looking for the break even interval on a graph of monthly profits picked the flat interval because they were thinking break even literally meant to find the interval that is even.

Note that the phrase breaking even is not a formal mathematical term but rather a conversational idiom with mathematical implications. Pimm (1987) calls these types of phrases locutions- "certain whole expressions whose meanings cannot necessarily be understood merely by knowing the means of the individual words, that is, the expressions function as semantic units on their own" (p.88).

• How can teachers become more aware of the mathematical locutions embedded in their classroom conversations? Pre-teach all mathematical terms, especially if they are also used in a nonmathematical context? Think literally.
  

Scenario #6: The Wordsmiths

*Students related exponential decay to exponential growth but couldn't figure out the precise terminology: "undoubling, doubling down, divided in half, taking half of it, dividing by two, halving".

• Would this particular mathematical conversation have occurred if the students didn't have a graphic "prop"?
• How did the visual representation act as a catalyst for student discourse?
• How might conversations about a graphic display encourage students to put their mathematical perceptions into writing?
• Now that a conversation had begun on the topic of exponential decay, how might the teacher build on the richly descriptive terms that the students created to press for more precise mathematical terminology?
  

Scenario #7 Where's the Fourth Fourth?

*Benjamin folded a paper strip into fourths and wrote on each of the folds 1/4, 2/4, and 3/4. When asked what his strip created, he said thirds. He was judging each strip based on the last numerator instead of the denominator. When asked about his strip that was folded in half, he wanted to say halves but according to his system, it would be one whole, based on the one in the numerator of one-half.

• What distinction was Benjamin making between the terms thirds, three folds, and dividing in to three parts?
• How might students conceptualize fourths differently, depending on whether they are asked to label the strip's segments (1/4, 2/4, 3/4, 4/4) or its folds (1/4, 2/4, 3/4)?
• What consideration might the teacher need to give to precision of language when providing directions for this task?
• What role might peer-to-peer discourse have played in helping Benjamin test his conjecture?
• What questions might a teacher ask to check more deeply for understanding if, at first glance, a student's thoughtfully done work is apparently correct? Ask them how they would explain it to their grandma or ask them to create a new example?
  

Scenario #8: Three Three/Ten Combos

*Benjamin and his class were instructed to divide a square pan of brownies into 30 equal squares. Students drew 5 rows of 6, 3 rows of 5, and 2 rows of 15. Later they were given a sheet with a giant square divided into 10 vertical strips. When asked to model three-tenths, Benjamin drew three horizontal lines across the vertical strips instead of shading in 3 of the 10 strips.

• From what assumptions might Benjamin have been working when he approached this new concept?
• Where was there a language-based component in his confusion?
• What do the phrases "three-tenths" "three and tenths", and "thirds and tenths" mean mathematically? What do they seem to mean to Benjamin?
• In a roomful of 20 students, how difficult is it to hear the subtleties of different word forms?
• What might Benjamin's drawing imply about what he heard?
• How could the teacher use Benjamin's drawing to encourage him to express his personal understanding in words?
• How might making additional sketches have helped Benjamin communicate what he head and understood?
  

Scenario #9: Holes in Her Logic
*Sarah was dividing donated food into boxes for 24 families. She drew 24 squares and drew dots in each box until she ran out so that they would be evenly distributed. For 12 pounds of cheddar cheese, Sarah divided 12 by 24 in the calculator and got .5, so she then drew 5 dots in each box instead of half of a dot.

• What windows to Sarah's thinking do her cocoa, milk, and cheese drawings provide?
• How did Sarah seem to understand division in general? What about dividing in situations when the quotient is less than one?
• What did Sarah believe about the decimal point?
• What was Sarah hearing, and how did this influence the manner in which she conceptualized decimal fractions?
• What is appropriate calculator use for students beginning to work with fractions and decimals?
• Would language-based cues, such as the teacher's asking how to divide a 12-pound wheel or chunk of cheese among 24 families, suggest useful visual models? Or might metaphors create further confusion? If I was Sarah and the words wheel or chunk were mentioned, I would have drawn wheels or chunks instead of dots but I still would have drawn 5 wheels or chunks in each box.
• What might the teacher do ti reinforce the use of proper terminology when students are working with decimal fractions?
  

I don't feel like typing Scenarios #5 and #10.


When teachers are asked to reflect on how student drawings can inform their practice, three themes emerge. Teachers feel that the drawings:

1. Make the students more aware that they're "speaking mathematics" in class,
2. Show a need for greater precision in the students' use of mathematical language, and 
3. Suggest areas in which directions and explanations should be more clearly phrased.
   

Suggestions from Teachers

• Combine verbal with visual.
• Monitor your language for words with double meanings.
• Assume positive intent to understand even from silly questions or offhand remarks.
• Ask students to move from drawing detailed pictures to simpler shapes, another step toward abstraction.
• Consider the sequence of representations students select. Does it promote depth of mathematical thinking?
• Actively point out connections to other representations so that students become fluent in translation from one representation to another.
  
  

 Suggested Practices

• Articulate and enunciate!
• Keep writing utensils available if you expect students to draw.
• Extra time: putting ideas on paper takes more time than talkingg.
• For open-ended questions, suggest drawing a diagram first in order to have something concrete to write about.
• Make copies of important textbook pages so that students can highlight, draw on, underline, and write notes in the margin to better interact with the text.
• Have students use graphic organizers that incorporate both words and pictures. 
• Designate a section of notes/worksheets for drawing, showing that you value and expect to see visual thinking.

Drawing is a device to capture the language of mathematics in order to make it visible to themselves.

Drawing slows students down and allows them to self-correct their thoughts while their hands are sketching; it also helps them to keep track of and record their solutions.

*Paraphrased