Swimming Against the Tide – The Dilemma of

Contextual MathBio-Link Summer Fellows

June, 2012Lisa Seidman and Jeanette Mowery

We Have a Problem with Math

Story of Max

• Max has a MS in biotechnology.• He was a student in our stem cell

technology program.• He messed up his cells because of

calculation errors.

LABORATORY MANUAL1. Prepare two 6-well culture plates with Matrigel basement

matrix.2. Prepare 6 mL of a working solution of 0.02% EDTA from a

stock solution of 2% EDTA.3. Determine the total number of wells that you will be

passaging into and the required volume of Matrigel. The final plating volume of each well of a 6-well plate should be 2 mL.

4. Examine the culture to be passaged and remove any differentiation if necessary.

Notes: Matrigel is required for culturing of Stem Cells in our system. Matrigel is stored at -80C in 1 mg aliquots. It remains frozen at -20C to -80C, is liquid at 4C, and gels rapidly at room

temperature. One aliquot is sufficient to treat two 6-well plates.

First Math Issue

• Conventional dilution problem• Concentrated stock solution• Need to dilute it to the correct final

concentration

LABORATORY MANUAL1. Prepare two 6-well culture plates with Matrigel

basement matrix.

2. Prepare 6 mL of a working solution of 0.02% EDTA from a stock solution of 2% EDTA.

3. Determine the total number of wells that you will be passaging into and the required volume of Matrigel. The final plating volume of each well of a 6-well plate should be 2 mL.

4. Examine the culture to be passaged and remove any differentiation if necessary.

C1V1 = C2V2

Step 1 - Define the variables:C1 = 2% V1 = ?

C2 = 0.02% V2 = 6 mL

Step 2 – Solve for V1:

(2%)(V1) = (0.02%)(6 mL)

V1 = 0.06 mL, or 60 microliters

Therefore, 60 microliters of 2% EDTA in a total volume of 6 mL of PBS equals a working solution of 0.02% EDTA.

Max

• Did not have a strategy for working with dilution problems.

Matrigel

Matrigel is required for culturing of Stem Cells in our system. Matrigel is stored at -80C in 1 mg aliquots. One aliquot is sufficient to treat two 6-well plates.

(1) What is the total volume you need to resuspend the 1 mg aliquot of Matrigel?

(2) What is the concentration of Matrigel and how much Matrigelis in each well?

Given: you have a 1 mg aliquot of matrigel that will be applied to two 6-well plates. Each well of the 6-well plate will need 2 mL of media.

Answer:(2 plates)(6 wells/plate)(2 mL media/well)

= ? mL mediaCross cancel units, and solve to get 24

mL of media

What is the Concentration of Matrigel and How Much Matrigel in

Each Well? Given: You have a concentration of 1 mg Matrigel in 24 mL.

Answer:1. Use proportions to determine the concentration of Matrigel:

(1 mg/24 mL) = (x mg/ 1 mL)Solving for x, x = 0.041 mg/mL, or 41 micrograms/mL

2. Knowing that there are 2 mL per well, we can use proportions to determine the amount of Matrigel in each well:

(41 micrograms/mL) = (x micrograms/2 mL)

Therefore, there are 82 ug of Matrigel per well of a six well plate.

The Problem

• Students, not just Max, can’t do the calculations they need.

• This leads to:– Frustration– Slowing pace of classes– Expense due to mistakes– Attrition– Failure

• In real world can have more severe consequences– We have heard of BS grads who were fired from a

company because of this problem

So, We Try to Solve the Problem

• Least successful method is to send students off to take math classes

• Might have math modules in courses• Bridge courses• Might have separate laboratory

calculations course – best option• Maybe these strategies solve the

problem

ROOT CAUSE

• Learn from the quality experts that it is not enough to identify problem

• Not enough to solve a problem• Need to identify and fix root cause

– otherwise problem is likely to recur

Tend to Think Root Cause is Lack of Math Skill

• Blame previous teachers • Blame the student for not paying

attention

But What Math is Required?

• Multiplication• Solving equation, simple algebra• Proportions• Max’s does NOT have problems

with the math• In fact, very few of our students

have problems with the math.

• Why did he have problems?– Strategy missing– Couldn’t discern the math required

amidst the other issues– Kinesthetic – Background

This Means

• Actual math required to do most calculations is within ability of average students

• The problem for students is finding the math when it is disguised within a context

This Means

• Root problem is not really math deficit• It is problem with finding the strategy

when it is disguised in a context

Contextual Math Has Certain Qualities

• Has language, materials, methods that may be unfamiliar

• Calculations may be difficult to find

Contextual Math

• Has consequences– Mars lander – that missed Mars– Deaths due to drug overdoses

It Also Means that

• Contextual math is NOT remedial or developmental math

CONTEXTUAL MATH ≠ REMEDIAL MATH

This is Why

• Once we accept this, we know that sending students off to take traditional math classes won’t help

• Almost every student, regardless of background, benefits from instruction in biotechnology math– This includes students with Bachelor’s degrees– Students with calculus background– For this reason, we require laboratory math

course for all students, even post-bacs and students who have had calculus

Have We Solved The Problem?

• Perhaps by adding a contextual lab math class we have identified and solved the root problem

So, Is there a Problem?

• Yes, an even BIGGER Problem

• People cannot solve problems in any practical context

• Not just biotech– Health professions– IT– Trades– Business – Etc.

• Therefore, all have specialty math courses that are contextual

Why this Problem?

• Maybe the root problem is that the academic community does not value contextual math

• Such math is considered to be “developmental”

• Fear of “dumbing down” curriculum• Therefore our students have not

learned to apply the tools they learn in math classes

Assumption

• Idea that students will learn abstract concepts first and then apply broadly to different applications

• Also think contextual problems are “boring”

Assumption is Incorrect

• Remember Max? It didn’t work for him.

• Doesn’t work for many (most?) students

“You do not study mathematics because it helps you build a bridge. You study mathematics because it is the poetry of the universe. It’s beauty transcends mere things.” Jonathan David Farley in NYT response

Common Core Math Standards

• This chasm is reflected in these standards

• Adopted by 40 states

Measurements and Data– Finished by Grade 5

• Measure lengths indirectly and by iterating length units.• Represent and interpret data.• Measure and estimate lengths in standard units.

• Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

• Geometric measurement: understand concepts of area and relate area to multiplication and to addition

• Convert like measurement units within a given measurement system.

• Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Algebraic Expressions and Equations by Grade 8

• Reason about and solve one-variable equations and inequalities.

• Represent and analyze quantitative relationships between dependent and independent variables.

• Use properties of operations to generate equivalent expressions.

• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

• Understand the connections between proportional relationships, lines, and linear equations.

• Analyze and solve linear equations and pairs simultaneous linear equations.

• According to standards, by grade 8, have learned almost all math tools needed for majority of occupations

• But do they ever learn how to use them?

What are They Learning in High School?

• A-APR.2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

• A-APR.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

• Use polynomial identities to solve problems.

• A-APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

• A-APR.5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

TAKE HOME MESSAGES

• Contextual math ≠ remedial math– Contextual math is not “dumbed down”– Most students struggle with contextual

problems

• It is alright to spend time on contextual problems where the math is not “difficult”

• If the math community won’t teach math that students need, we need to do it ourselves

Sol Garfunkel and David Mumford

Op Ed in NYT“How often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a ‘group of transformations’ or a complex number?...A math curriculum that focused on real-life problems would still expose students to the abstract tools of mathematics…But there is a world of difference between teaching ‘pure’ math, with no context, and teaching relevant problems that would lead students to understand how a mathematical formula…clarifies real-world situations.”

They Conclude With:

“It is through real-life applications that mathematics emerged in the past, has flourished for centuries, and connects to our culture now.”

“In 1953 you were my math teacher. You promised me that polynomials would come in handy someday. How much longer do I have to wait?”