Teaching Middle School Students:

Caitlin Thomas

Making Math RelevantEnjoyableAccessibleLearnable

Discussion

• A long time ago, in a galaxy far, far away… we were once all middle school math students

•What were your classes like?•What were your teachers like?•What kinds of things did you do in your

classroom?

Why do we teach math?• To teach students:

– The basic skills of math (how to do arithmetic, etc.)– How to think logically– How to handle abstractions (the underlying concepts that guide real

world applications)– The good mental habits for solving any kind of problem

• Read the question/evaluate the situation• Identify all of the relevant information you have been given• Establish what you are trying to achieve• Know and understand how to use your tools• Combine the information and your given tools in a way that will result in

achieving your goal

But what are our students actually learning?

The International Problem

• U.S. ranks near the bottom in both math and Science• Ranking has been falling since 2000• In 2006, US ranked 25 out of 30 in Problem Solving Abilities

Source: Organization for Economic Cooperation and Development (OECD), PISA 2006 Results, http://www.oecd.org/

Rankings are for the 26 OECD countries participating in PISA in 2000, 2003, and 2006.

Subject 2000 Rank(out of 26)

2003 Rank(out of 26)

2006 Rank(out of 26)

Mathematics 17th 22nd 22nd

Science 13th Tied for 17th 19th

The Domestic Problem• Achievement Gap:– Low-income and minority 8th graders score 2-3 grade levels

behind their higher income, predominantly white peers in math and science

Source: NAEP, Long-term Black math scores. http://nces.ed.gov/nationsreportcard/ltt/results2004/natsubgroups.asp

The Individual Problem

• When tutoring students, talking to people who don’t “do” math, etc. what are some common things you hear?– “I can’t do it, and I don’t care.”– “It has nothing to do with my life.”– “Math is hard.”– “I give up.”– “I hate math!”

Why do people think this way?• Math in American culture– Math talent is innate

• If one is not naturally good at math, there is little one can do to become good at it.

– Math is hard!• Commonly found sociological phenomena• Generally in popular culture (TV, movies, etc.)• 1992, Teen Talk Barbie dolls used to say “Math class is hard”

– Talent in math is not valued or encouraged as much as in other countries.• The winners of rigorous math competitions are predominantly

immigrants or children of immigrants from countries that highly value math education and math talent.

Consequences of this view: Math Anxiety

• Math anxiety: the panic, helplessness, paralysis, and mental disorganization that arises among some people when they are required to solve a mathematical problem.

• Causes: – Preconceived notion that math is hard or impossible– Teaching techniques that emphasize one RIGHT answer or

one RIGHT method– Memorization in place of understanding

Consequences of Math Anxiety• Mental disorganization, incoherent thinking• Negative self-talk• Inability to recall material studied • As math anxiety increases, math achievement decreases• Tendency to speed through problems, sacrificing accuracy• Underdeveloped numerical sense making abilities• The preoccupation with one’s dislike or fear of math

distracts thoughts and interrupts the actual learning process

• AVOIDANCE

What should we do?

• Imperative that we close the achievement gap and increase the competitiveness of American students internationally

• Make math REAL– Relatable– Enjoyable– Accessible– Learnable

Two Basic Approaches

• Two basic approaches to teaching math:

Computational

Approach

Conceptual

Approach

Computational Approach

• Computational-procedural approach to teaching math– Routine drill and practice– Focus on short term memorization– Results in the ability of students to apply their

knowledge only to similar situations

Computational Approach

• In a completely or mostly computational/procedural approach

• Memorization of formulas• Plug in and solve

Find the volume of each solid to the nearest tenth. (use pi = 3.14)

1.  rectangular prism: l = 14 ft, w = 20.6 ft, h = 10 ft 2.  sphere: r = 21 yd

3.  cylinder: radius = 2.1 cm, height = 4 cm 4.  cylinder: d = 2 in, h = 21 in

5.  pyramid: B = 27 km2, h = 5 km 6.  cylinder: r = 9 m, h = 15 m

Math Transfer

• Weaker “transfer” when students learn only through a computational approach that emphasizes memorization over application

• Transfer is the ability to take one learned concept or skill and apply it in a different situation or to a different problem

Conceptual Approach

• Conceptual approach to teaching math– Emphasis on real world problem solving and

student reflection– Working on problems with no obvious solution– Using investigation to solve problems

Conceptual Approach• Project:

– Design a model (on paper) of a product that you are going to manufacture (action figure, doll, piggy bank, etc.).

– Record the measurements of the product (height, width, depth) on your design.– Design a box which will serve as packaging for your product (your product must fit in the

box!).– Determine the cost of manufacturing your box by using the following pricing chart:

– At least one face of your box should be made of clear plastic in order to see the product.– All products cost $10 to manufacture. Taking into consideration the price of your packaging,

determine how much you should sell your product for. Calculate the profit you will generate from selling 500 units of your product.

Material Price Per Square Centimeter

Cardboard $0.05

Clear Plastic $0.07

Activity

•Which activity was more typical of a computational approach? Why?•Which activity was more typical of a conceptual approach? Why?•What skills were being focused on in the first activity ? The second?•Which activity did you prefer? Which would students prefer? Why?•What do you think are some of the benefits of the project style activity ? Or the benefits of the more traditional style worksheet?

Teaching Math in the United States

• US math teachers are traditionally more likely to use the computational/procedural approach– Lecture, students take notes, worksheets and drills

• Teacher education programs focus more on the computational approach

Teaching Math in the United States

• Studies show that a blend of both approaches is used in the US, but:– Students receive significantly more computational

teaching than conceptual teaching– High achieving students receive greater exposure

to conceptual teaching– Struggling students receive more computational

teaching—depresses already low math abilities even more

Teaching Math in Other Countries

• In many of the countries (Japan and Germany) that outperform the United States on international measures of math achievement:– More use of conceptual methods (integration)– Students explore, investigate, and solve problems

with greater insight– Result in greater problem solving abilities

Computational-Conceptual Disconnect

• Ability to “make sense” of math is to draw the connection between procedural/computational math and conceptual math

• Problem on NAEP 8th Grade Assessment:

An army bus holds 36 soldiers. If 1,128 soldiers are being bussed to a training site, how many busses are needed?

• Only 24% of national sample of 8th graders solved this problem correctly

Computational-Conceptual Disconnect

• Student Thought Process Leading to Unsuccessful Solution

Story Situation

Mathematical Model:

Computation

Story Text

Computational-Conceptual Disconnect

• Student Thought Process Leading to Successful Solution

Story Situation

Mathematical Model:

Computation

Story Text

Importance of Making a Connection

• Computational accuracy with conceptual understanding

• Student performance is adversely affected by dissociation between sense making of solution and completion of the problem

• Stronger connections between school math problems and real world situations

• Need to practice written explanations and interpreting solutions

How to Make the Connection in the Classroom?

• One solution:Math in Context (MiC) curriculum

• Research has show MiC tobe effective in developingstudents’ math power

What is MiC?

Real World

Problems

Abstract Concept

s

Apply to New

Situations

What is MiC?

• Does not focus on one skill or domain at a time

• Students explore the relationships between different domains or skill sets

• Encourages students to collaborate to solve problems

• Values creativity and encourages multiple ways of solving problems

What is MiC?

• Reinventing significant mathematics• For example: area of a triangle

• Students learn when they discover for themselves

• Knowledge has to be built by each student in his or her mind

Cereal Numbers

• Goal: 7th graders to learn various ways of multiplying and dividing fractions

• How would you do this?• How did your middle school teacher do this?• How does MiC do this?• By using the context of producing, tasting, and testing

various cereals

Cereal Numbers

Sample question:

How many cups of Corn Crunch #1 are needed for 60 taste testers if the serving size of the cereal is 1 ½ cups and each test portion is one fourth of the serving size?

Cereal Numbers

Student Responses#1: 22 ½ cups of cereal. What I did, I went from four by

four. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, and I counted them and they were 15, and so I multiplied 1 1/2 which is for 4 people times 15 and got 22 ½.

#2: 22.5. We found the answer by multiplying 1 ½ by 60 because there are 1 ½ cups in Corn Crunch 1 and 60 people, then I divided it by 4 because there are 4

servings in 1 ½ cups and what I want to know is how many cups (4 servings) are needed for 60 servings.

Cereal NumbersStudent Responses#3: If 1 ½ cups were for 4 people, then we need to know how much

serving per person, because if you multiply 1 ½ x 60, it won’t work. So we divided 1 ½ by 4 people, then we already know each person’s serving. Now we multiply by 60, so we could find the answer. 1 ½ / 4 x 60 = 22 ½ (22.5) cups.

#1: Count by fours and then count how many 4’s go into 60 and then multiply by 1 ½ (60 / 4) x 1 ½

#2: Found how many cups for full 60 servings and then divided by 4 (1 ½ x 60) / 4

#3: Found size of test serving, then multiplied by 60(1 ½ / 4) x 60

Cereal Numbers

• The teacher did not present one particular way to solve this problem

• Instead students invented their own solutions, clearly communicated most work and thinking, and provided rationale for their choices

Cereal Numbers

• Students begin to see the math skills as tools, even if they are difficult

• Doing this helps students learn that math is real

• Importance of justifying explanations• Puzzle nature of the way problems are

presented make them more exciting and engaging

Teaching Math for Social Justice

• Eric Gustein• Goals:– For students to begin to read the world

(understand complex issues involving justice and equity) using math

– For students to develop mathematical power– For students to change their orientation towards

math

Teaching Math for Social Justice

• Takes the real world examples beyond just the math

• Makes the clear connection that math is a TOOL that can be used to analyze and solve problems in the real world

• For example: use racially disaggregated data on traffic stops to learn about concepts of proportionality and expected value

Housing Prices Project

• Gave students data for highest median house price in a nearby suburb (McFadden, $725,250)

Housing Prices Project• How could you use mathematics to help answer whether

racism has anything to do with the house prices in McFadden. Be detailed and specific!! Describe:– What mathematics would you use to answer that question?– How would you use the mathematics?– If you could collect any data to answer the question, explain what

data you would collect and why.– Give examples of data that would cause you to believe that racism

is involved in the McFadden housing price, and explain why you reached that conclusion based on the analysis of the data.

– Give examples of data that would cause you to believe that racism is not involved in the McFadden housing price, and explain why you reached that conclusion based on the analysis of the data.

Housing Prices Project

• Student response:“First I would find the price of a house in McFadden and a working-class Latino’s annual income. Then I would divide the house’s price to find the annual payment. Then I would take the income and subtract the cost of living (food, clothes, etc.) and compare the different to the house’s payments. If there is a large difference in the two, it is possible discrimination exists in the prices of McFadden because the price is impossible for most, if not all, working-class Latinos. I would say there was no racism involved if the prices were more affordable for minority groups.”

So… what is the main point?

• http://www.youtube.com/watch?v=vFRTgr7MfWw&feature=fvw

• “No, no, no. I was a good student, I just couldn’t get into math—couldn’t see how it could relate to the real world.”

• kagulya (7 months ago): i wish my math teacher was like charlie, math wouldn't be as boring as it is now

Discussion

• What does this mean for math teachers?– Must be highly educated, enthusiastic, and

passionate about teaching math– Must communicate this excitement and energy to

their students– Must have the in-depth knowledge to create

engaging lesson plans that help their students to relate math to the real world

– Must convey the creativity and beauty that too many people do not see in math

Discussion

• What does this mean for math teachers?• What does this mean for math curriculum?– Must place more emphasis on the conceptual

approach to teaching math– Puts the teacher in the redefined role as a guide

and a facilitator of inquiry– Developing our students as critical problem

solvers

Discussion

• What about No Child Left Behind, accountability measures, and standardized testing? (the “teaching to the test” phenomenon)

• What are the benefits to a MiC approach? • What are some obstacles to the MiC approach?• How would your middle school math class have

been different if it had been taught in a more conceptual way?

Homework

• Complete activities on the homework assignment sheet

• Evaluation

References• http://mathematicallysane.com/analysis/polya.asp, The Goals of Mathematical

Education, George Polya• Economic Impact of the Achievement Gap in American Schools, McKinney&Company,

2009• National Center for Education Statistics, National Assessment of Educational Progress,

2007, Mathematics Assessment, 2005 Science Assessment• Math Skill Suffer in US, Study Finds. S. Rimer, Oct 10, 2008, New York Times• Math Anxiety: Personal, Educational, and Cognitive Consequences, MH Ashcraft, Current

Directions in Psychological Science, Vol. 11, No. 5 (October 2002), pg. 181-185• Assessing Barriers to the Reform of US Mathematics Instruction from and International

Perspective, Desmoine, LM, Smith T., Buler D, Ueno K, American Education Research Journal, Vol. 42 No. 3 Autumn 2005 pgs. 501-535

• Sense making and the solution of Division Problems Involving Remainders: An examination of Middle School Students’ Solution Processes and their Interpretation of Solutions; Silver EA, Shapiro LJ, Deutsch, A; Journal of Research in Mathematics Education, Vol. 24, No. 2, 1993 pg. 117-135

References• Smith, M. S., & Silver, E. A. (1991, April). Examination of

middle school students' posing, solving and interpreting of a division story problem. Paper presented at the annual meeting of the American Educational Research Association, Chicago.

• Math Anxiety: Myth or Monster?; Kay Haralson, Powerpoint• http://www.mmmproject.org/number.htm• Teaching and Learning Mathematics for Social Justice in an

Urban, Latino School; Eric Gustein; Sournal for Research in Mathematics Education, Vol. 34, No. 1 Jan, 2003, p. 37-73