Chapter 5 - Solution Methods and Problem Solving Strategies Classroom Discussions: Using Math Talk to Help Students Learn ...let's revisit the last time we met. What kinds of misconceptions and errors do we want students to discuss and explore?

What kinds of errors would we rather not discuss? For the next time we meet... Thank you!!!

Brought to you by the Grade 2 team! Patty Milewski

Beth Krow

Dana Finberg

Laura Botsis

Special Guest Justine Gannon

Problem-Solving Strategies

•When there is no immediate solution to the problem

•Requires discussions about solution methods and problem-solving strategies

•This can reveal holes in understanding, misconceptions, and overgeneralizations that impede learning

Solution Methods

•A procedure which the solver makes a plan for reaching the solution and carries out the plan

•George Polya’s 4-Step Problem-Solving Method: describes a process of mathematical thinking. The solver moves back and forth among the steps until they’re solved.

1.Understand the problem

2.Make a plan for solving the problem

3.Carry out the plan; and

4.Look back and reflect on the answer in terms of the initial question Problem Solving Strategies and Representations Talking Through the Givens of a Mathematics Problem Good problem solvers:

•Spend a lot of time understanding the problem and all relevant relationships.

•Ask themselves key questions about the given and unknown information.

•Monitor their own progress.

•Reevaluate choices and sequences.

•Find and correct errors along the way. Ms. Dunbar

4th Grade Teacher •Before solving the problem, you need to understand the information in the problem and what the question is asking.

•Read the problem to self (twice).

•Tell some information about the problem.

•Understand all the facts and ideas in the problem first.

•Where did you get that information?

•Records relevant facts on board.

•State in your own words what question your trying to answer. What to do when a student is unable to answer a question. •Ask for someone to repeat the question, repeat an answer, or repeat some information.

•Immediately return to the confused student and ask them to repeat what the student had just said. Talking Through a Method for Solution Back to Ms. Dunbar-

She asks students to work in pairs to solve the problem. She notices that most students use multiplication and division, but one pair uses repeated addition. Some students make computational errors when dividing. Ms. Dunbar decides to use a whole class discussion of the problem to address these issues. Ms Dunbar asked students who were confused to revoice and explain other students’ explanations of their effective methods. She also asked students to predict what came next in solving the problem. When she did this she checked back with the original speakers to see whether the prediction was accurate. Many students who had not solved the problem or who had made mistakes were able to make progress in their understanding of the problem and that solution method. Increasing students' self monitoring of their thinking Assessing Student Learning “Representing ideas and connecting the representations to

mathematics lies at the heart of understanding mathematics” (90).

A Plan as a Strategy - Involves representing a mathematical idea in a different form

Commonly used Problem-Solving Strategies - Guessing, Checking, Making a list or table, Looking for a pattern, Using an algorithm, Model, Draw

Well Known Representations: + - x ÷ and line graphs

Students should be able to understand these and connect them to other conventional representations:

•division can also be represented with fractional notation, 8÷2 or 8/2, pictures, and objects

•the points on a line graph that satisfy the equation can be

show in a table of (x, y) values Introduction Doug uses X’s and marks in a nonstandard way to keep track of the number of 10’s and 1’s.

Doug, a 2nd grader, adds a column of 2-digit numbers by first looking at the 10’s column and recording one X on his paper for each group of 10. He then counts the number of X’s and records that numeral in the 10’s column. He next makes marks for the 1’s and counts them before recording the numeral. Sometimes he has to adjust the numeral he writes in the 10’s column if there are enough 1’s to make another 10 (90). Non-standard Representation “Classroom talk can help students

transform their understanding of that representation

and its potential.”

Classroom talk can enhance students understanding.

A teacher uses a discussion about a problem on addends to highlight students’ problem-solving strategies and help students link different representations.

Mr. Evans’ 1st grade students are discussing a problem about 10 students who are going to an amusement park with Mr. Evans and Mrs. Zito. The children have to be with one of the teachers at all times. How many children can be in each of the 2 groups?

•Partner talk

“Talk to your partner about what Robin and David’s picture means.”

•Restating/Revoicing

“I’d like Pedro and Briana and Suman and Josh to come up together and tell us in their own words about Robin and David’s drawing.”

•Follow-up question

“How do you know which circles go together to make ten?” Classroom Talk Connections for Teachers * “Our goal is for students to be able to understand and explain the mathematical meaning of each part of their symbolic, graphic, or language-based representation. Furthermore, we want them to be able to understand and explain the connections among different representations” (91).

* “As teachers, we sometimes take it for granted that students will understand various forms of representation in the same ways that we do.” Connections for English Language Learners Teachers should provide ELL’s with opportunities to respond

to challenging questions through response formats appropriate to these

students’ oral proficiency levels such as:

yes/no, either/or, short answer, or extended response options

Another technique that is helpful in supporting ELL’s learning in academic content areas is frontloading, or, teaching the text backwards.

•Identify important concepts, vocabulary, and questions before a lesson

•Elicit and link student’s background knowledge

•Provide hands-on experiences that invite key questions

Identify language demands of the content that may be particularly challenging for ELL’s.

For example, the vocabulary of math has specialized terms like equation and denominator, but it also has specialized use of common terms such as table, column, and round. ELL’s may have learned meanings for these terms that do not apply to math.

Lastly, although making content comprehensible through visual aids and hands-on experiences is important, they need to move beyond strategies that help ELL’s “get around” language to include teaching academic language.

From Misconceptions about Teaching English Language Learners Discussing methods of solution and problem solving

strategies also helps students learn more sophisticated

problem solving methods.

Mrs. Steinfield showed her class a picture of red and blue marbles in a jar. She asked her class how many more marbles she should put in the jar and what color they should be so that there will be three times as many blue marbles as red ones. Many students revealed that they had to use a drawing to find the answer. Mrs. Steinfeid called on a student Bora to explain the answer. She also called on several class mates to explain Bora’s method as a way to assess their understanding of her explanation. Bora had to talk about her reasoning a few times until most students understood it. She also asked others to explain why Bora’s method was reasonable and efficient. Mrs. Steinfeild gave the class two similar problems and encouraged them to use Bora’s method. She didn’t assume students would use the method after hearing it so had them explain Bora’s solution method and apply it to new problems. Analyzing student work and gathering data during classroom

discussions are two ways to uncover patterns of errors and misconceptions.

In Mrs. R’s class, discussion was held as to how the students solved a simple comparison subtraction problem where two numbers are being compared.

Mrs. R discovered that students did not know that they could solve comparison number stories using subtraction. Rather than use subtraction, most of the students drew pictures and used counters to compare the two groups.

The misconception was that the students thought that using subtraction to solve the problem wouldn’t work because nothing was being taken away.

The discussion continued with Mrs. R. posing questions to the class in order to get them to talk about their strategies. This gives teachers the opportunity to assess students’ understanding of content applied to situations.

After the discussion, Mrs. R decided to focus on the vocabulary of comparison number stories and use more problems involving number

comparisons in her lessons. Talking about solving problems develops awareness of a students' own understanding.

This leads to better self-monitoring and reflection.

Self-monitoring can also be referred to as "metacognition".

The process of explaining oneself and answering probing questions builds metacognitive strengths.

The goal is for students to eventually monitor their own

thinking independently. What does this look like in the classroom? During a classroom discussion, Mr. Cooper asked students:

How did you solve this problem?

How did you check your work?

Who would like to respond to this student's work?

Does anyone have advice for a student who's struggling?

Instead of using discussion to reveal mistakes, Mr. Cooper used the discussion to support thoughtful reflection. Sometimes a student whose confidence is quite fragile will present a solution method that is deeply flawed.

What can we do as teachers to help the class see that discussing different solution methods, right or wrong, helps move everyone toward understanding mathematical truth?

How do we do this and be sensitive to the individual needs of students?

Page 108, question 3 Let's take a look at a 3rd Grade classroom. After the video, we're going to ask you to discuss with your group the following questions:

What was the purpose of this discussion?

How do you think it would benefit your students to do something similar prior to working on a difficult problem? Exit Slip:

Write down one or two things that stuck on you from the presentation.

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# Classroom Discussions: Using Math Talk to Help Students Learn - Chapter 5

Methods for teachers to encourage students to talk about math. Talking about math can truly enable a student to develop deeper understanding.

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