# Presentation: 037073

## Looking at Student Work toFocus Secondary Math Instruction

- Rhonda Naylor
- Archived Informaiton

**Notes:**
 Read an excerpt from Score One for the Sloths, by Helen Lester, 2001, Houghton Mifflin, through p. 23.

 With No Child Left Behind, we may find ourselves worrying about the “SOS”.  But don’t worry:  our workshop today is designed to focus on math standards, deepen our content knowledge in the field of algebra, enhance pedagogy and strength our classroom-based assessment skills, with collaboration of colleagues, to focus our mathematics teaching, and develop our skills for examining results to make decisions about instruction in our classes.

I am a classroom teacher of 8th grade math, teaching any combination of Pre Algebra, Advanced Algebra, Integrated Algebra and Geometry, and Advanced Geometry each year.  Let’s find out about each of you.  Who teaches middle school?  High School?  What else brings you here?  

4.  Today we will look at student work, a powerful way to collaborate to focus on evaluating student learning in relation to a standard.  

5.  We will complete a task, we will unpack the algebra standard related to patterns and algebra and how these standards relate to the task.  We will then examine 6 pieces of student work of varying quality.  We will discuss which pieces show the conceptual understanding needed to meet which elements of the standards and why.   We will discuss using rubrics to score student work. We will talk about the instruction each student is ready for next, and what it might look like.  And how rich student work helps us evaluate progress, and make instructional decisions.  

## Before It’s Too Late

**A Report to the Nation **
**From the *****National Commission on Mathematics and Science Teaching for the 21st Century***
**After an extensive, in-depth review of what is happening in our classrooms, the Commission has concluded that the most powerful instrument for change, and therefore the place to begin, lies at the very core of education—**
**with teaching itself.**

**Notes:**
A report from John Glenn’s commission set the stage for workshops like this one.   

## Before It’s Too Late

**Specific teaching skills—for example, the ability to distinguish between what is most important for students to learn and what is hardest for them to understand—can only be acquired through training, mentoring, collaboration with peers, and practice.**

**Notes:**
We hope that you gain expertise through collaboration today, and continue that collaboration when you return to your classroom in the fall.

## Why look at student work?

**To make classroom decisions that support student learning, teachers must understand both the mathematics and their students, and they must continue to grow in their understanding.**
- To have a deep understanding of mathematics.
- To have a deep understanding of the standards.
- To know the students well, to make instructional decisions to further their learning.
*	**From National Board for Professional Teaching Standards*

**Notes:**
We are all familiar with the standards for our students’ learning; here is an excerpt from standards for teachers, from NBPTS Standard III AYA mathematics Standards.  This says it all, doesn’t it?
Good classroom judgment--knowing when or whether to intervene, deciding which avenue to follow, knowing when to bring closure and move on- depends on teachers’ sensitivity to student’s learning, but also on profound mathematical knowledge that allows us to recognize and pursue mathematically significant ideas.  An increased emphasis on applications also puts new demands on teachers’ mathematical knowledge.  Sometimes we enter unfamiliar mathematical territory.  Time for collaboration, unpacking the standards first, so that T knowledge is reinforced. 

## Examine the Algebra Standards

**Algebra is a language for modeling problem situations, and for reasoning and drawing inferences about functions and relations.  **
**Interplay among numerical, symbolic, verbal, and graphical representations of quantitative relationships.**
**Processes:**
- Discovering, describing and reasoning about patterns represented in visual, numerical, and symbolic form
*	**	**	**NBPTS AYA Mathematics Standards*

**Notes:**
NBPTS Standard III AYA Mathematics Standards.

Unpack the standards:  look at verbs, nouns.  Students will understand that patterns model real world problem situations:

## Algebra Standards

**In the vision of school mathematics, middle-grade students will learn that patterns can be represented and analyzed mathematically.**
**By the ninth grade, they will have represented linear functions with tables, graphs, verbal rules, and symbolic rules and worked with and interpreted these representations. **

**Notes:**
Algebra is an extremely effective tool for investigating all areas of mathematics.  Algebra is a language of process and change.  (Navigating through Algebra in Grades 9-12, NCTM, 2001).

## Unpacking Content Standards

**	****Understandings and “Big Ideas”**
**Students will understand that...**
**Students will be able to ...**

**Notes:**
15 minutes
We want to “Unpack the standards” to drive big ideas, key knowledge and skills.  You may find templates from Understanding by Design, by Grant Wiggins and Jay McTighe, which guided our steps here. (2001, 2002, 2003, ASCD). 

We are going to examine the NCTM Algebra Standard by examining nouns and verbs, 
U:  using nouns, complete:  students will understand that...  What will students need to understand in order to really learn this content?
K: using verbs, and be able to state:  students will be able to.... What should students eventually be able to do as a result of such knowledge and skills?

## Algebra Standards

**In grades 9–12 all students should:**
**Understand patterns, relations, and functions**
**generalize patterns; **
**understand relations and functions and select, convert flexibly among, and use various representations for them;**
**analyze functions of one variable by investigating rates of change, intercepts, zeros;****	**
**Use mathematical models to represent and understand quantitative relationships**
**draw reasonable conclusions about a situation being modeled;**
**Analyze change in various contexts **
**approximate and interpret rates of change from graphical and numerical data.****                              ***NCTM*

**Notes:**
Wiggins suggests we examine the standard this way:
Highlight the verbs in yellow--(understand, generalize, select, convert, use, analyze, approximate, interpret)
Understandings (U):  Students will understand that ... (patterns and graphs model real world problem solving situations)

Continued on next page:

## Colorado Algebra Standards

**Students use algebraic methods to explore, model, and describe patterns and functions involving numbers, shapes, data, and graphs in problem-solving situations and communicate the reasoning used in solving these problems. **

**Notes:**
These are our Colorado Algebra Standards.  Very similar.  But notice the key words:  Communicate reasoning.  
The study of patterns, functions, and algebra helps learners to recognize and generalize patterns; identify and clarify functional relationships; and represent and manipulate these relationships verbally, numerically, symbolically, and graphically. Symbolic representation, including the many interpretations of the concept of variable, is important but only one of many ways to represent patterns and functions. Students who are adept at identifying and classifying patterns and functional relationships are better able to use these relationships in real situations, both in school and out. The portrayal of functions and algebra in this standard is broader, deeper, more connected, and more useful to learners than in the traditional high school algebra curriculum. 

## Colorado Algebra Standards

**In order to meet this standard, a student will **
** ****	**
**identify, describe, analyze, extend, and create a wide variety of patterns in data; ****	**
**describe patterns using mathematical language; **
**solve problems and model real-world situations using patterns and functions; **
**describe the connections among representations of patterns and functions, including words, tables, graphs, and symbols.**

**Notes:**
Notice the key words:  mathematical language, real-world situations.

Understandings (U):  Students will understand that ... (patterns and graphs model real world problem solving situations).

## The Task: Journey West

**Complete the task**

**Notes:**
Now we are ready to examine and complete our task, Journey West.  Read the intro.  Please work on the task now.
Walk around room, questioning, using PBS TeacherLine card, Developing Mathematical Thinking with Effective Questions.  Allow ample time for all to be familiar with all the questions, even if writing isn’t completed.
Once completed, discuss each of the questions in this task.
 Have participants share their approaches.   Goals:  be sure participants really understand the  math involved and think about different ways a student might approach the task.  How did you solve #3? 7 gallons first day, 14 gallons next 2 days, 14 next 2 days?  7gal/day * 20 days = 140 gallons.  Some did write an equation.  
What would I expect a student who is meeting the standard to do?
4.  What misconceptions do you feel students will have?

## Architecture of Accomplished Teaching

**Assessment of students**
- Who are the students and what do they need?
**Standards**
- Set high and worthwhile goals (standards) that are appropriate for these students at this time
**Instruction**
**Evaluate student learning on the standards**
**Reflect on student learning**
- Reflect on the instruction
- Make curricular decisions
*	**	**	**	**	**                   **NBPTS*

**Notes:**
From National Board for Professional Teaching Standards.  Link content (standards), instructional strategies, and assessment. Reflection about our student’s learning as a basis for instructional planning and decision making.  Strengthen classroom-based assessment skills to focus teaching (goal of these summer workshops)

## The Task:  Journey West

**Examine the standards in relationship to this task:**
- What knowledge will students be demonstrating with this task?
- What will students be able to do?
**As we look at student work, what is the most important feature that would be present in upper tier papers but not in lower tier papers?**
**What math language would be valued?**

**Notes:**
Knowledge:
Students will know y intercept has no meaning in this problem, but will know the meaning of the x intercept, when there is no water left.
Students will know rate of change is shown in a linear graph (how much water is consumed each day)

Skills:  students will be able to:
Plot points, converting from a table to a graph
Draw line of best fit
Interpret meaning of the intersection of lines
Extend a line or table (or determine an equation) to make predictions
Support statements with evidence from graph or chart
generalize patterns

## The 4 Point Rubric

**4 score points**
**All answers are correct **
**1) Smith, 9-11 gallons**
**2) A. 10 day     B. 15 gallons**
**3) 140 gallons**

**Graph is correct and correctly extends the lines to form a line of best fit.**
**Strong explanation in question 1 using sophisticated math language (Identifying the rate)**
**Justifications in questions 2 and 3 are based on data from the table and/or graph and use sophisticated math language or representations.**

**Notes:**
Handout:  The Rubric
Look at your handout of the Task Specific Rubric for Grade 10 Journey West.
(Okay, first you may look at your answers in relationship to the 4!)

## The 4 point Rubric

**3 score points**
**All answers are correct **
**1) Smith**
**2) A. 10 day     B. 15 gallons**
**3) 140 gallons or 130 gallons with justification**
**Graph is correct.**
**Strong explanation in question 1 using math language **
**Justifications in questions 2 and 3 are based on data from the table and/or graph.**

**Notes:**
Next, circle key words that distinguish a 4 from a 3.  

3:  correctness in skills; justifications based on table or graph, math language; strong explanation
4:  correctly extends line to form a line of best fit; sophisticated math language (ID rate) or representations

## The 4 point Rubric

**2 score points**
**Has two parts of the questions correct**
**Graph has the table data plotted correctly but does not extend the information**
**Some justification to questions 1,2,3, to the data and graph, but not complete**

**Notes:**
Circle key words that distinguish a 3 from a 2.
3:  correctness in skills; justifications based on table or graph, math language; strong explanation
2:  two parts correct out of 4; graphs correctly, but not extended; some justification, graph not complete.

## The 4 point Rubric

**1 score point**
**Has less than two questions correct**
**Graph is incorrect**
**Little to no justification to questions 1,2,3, by using the data and the graph**

## 
What does the student know?
What is the student able to do?
What are the next steps for instruction?
Will a note be enough?

**What does the student know?**
**What is the student able to do?**
**What are the next steps for instruction?**
- Will a note be enough?

**Looking at Student Work**

**Notes:**
Now let’s examine student work.  Let’s look at the 3 first.  

Divide work into upper tier (3,4) and lower tier (1,2).  When would a note be sufficient for helping the student reach the goal?  When would a note not be enough?

3:  This student is correct; student is able to plot data correctly, extend accurate lines of best fit.
Question 1 is correct with a strong explanation.
Questions 2a, b, c are all correct with justification based on the data from the graph.
Question 3’s reasoning supports another way to calculate the rate of consumption.

Instructional Opportunities:  ready to move onto more complex problems, continue having opportunities to write about their mathematical reasoning/problem solving to foster more sophisticated interpretation.  

## In Your Classroom

**Select a standard or objectives**
**Plan an instructional sequence**
**Select a performance task**
**Collaboratively examine student work**
**Reflect on what students learned **
**Reflect on instructional sequence and revise to ensure student learning**
- Share with us electronically

**Notes:**
Use this task or another of your choosing with your own class in the fall.  Post samples of your student work along with your comments on that student work.

## Resources

**Project Zero at Harvard Graduate School of Education**
**Annenberg Institute for School Reform at Brown University**
***Looking at Student Work For Teacher Learning, Teacher Community,and School Reform.***** Littie, Gearhart, Curry, And Kafka, **_**Phi Delta Kappan**_**; **
**Nov 2003**

## A Vision of the Future

**The star teachers of the twenty-first century will be those who work together to infuse the best ideas into standard practice.**
**They will be teachers who collaborate to build a system that has the goal of improving students’ learning in the average classroom, who work to gradually improve standard classroom practice...**
**The best we know becomes the standard way of doing something.**
**The star teachers of the twenty-first century will be teachers who work every day to improve teaching--not only their own but that of the whole profession.**
**	****	****	****	****	****        ***The Teaching Gap*

**Notes:**
Close with reading the end of the Teaching Gap.