Biotechnicians use math skills on the job that are not taught in math courses. Seidman describes the problem and discusses options for correcting it.

1. Swimming Against the Tide –
The Dilemma of Contextual
Math
Bio-Link Summer Fellows
June, 2012
Lisa Seidman and Jeanette Mowery

2. We Have a Problem with Math

3. 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.

4. LABORATORY MANUAL
1. 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.

5. First Math Issue
• Conventional dilution problem
• Concentrated stock solution
• Need to dilute it to the correct final
concentration

6. LABORATORY MANUAL
1. 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.

7. C 1V 1 = C 2V 2
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.

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

10. 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
Matrigel
is in each well?

11. 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 media
Cross cancel units, and solve to get 24 mL of
media

12. 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
• 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.

15. 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

16. 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

17. 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

18. Tend to Think Root Cause is Lack of
Math Skill
• Blame previous teachers
• Blame the student for not paying attention

19. 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.

20. • Why did he have problems?
– Strategy missing
– Couldn’t discern the math required amidst the
other issues
– Kinesthetic
– Background

21. 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

22. This Means
• Root problem is not really math deficit
• It is problem with finding the strategy when it is
disguised in a context

23. Contextual Math Has Certain
Qualities
• Has language, materials, methods that may be
unfamiliar
• Calculations may be difficult to find

24. Contextual Math
• Has consequences
– Mars lander – that missed Mars
– Deaths due to drug overdoses

25. It Also Means that
• Contextual math is NOT remedial or developmental
math
CONTEXTUAL MATH ≠ REMEDIAL MATH

26. 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

27. Have We Solved The Problem?
• Perhaps by adding a contextual lab math class
we have identified and solved the root
problem

28. 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

30. 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

31. Assumption
• Idea that students will learn abstract concepts
first and then apply broadly to different
applications
• Also think contextual problems are “boring”

32. Assumption is Incorrect
• Remember Max? It didn’t work for him.
• Doesn’t work for many (most?) students

33. “ 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

34. Common Core Math Standards
• This chasm is reflected in these standards
• Adopted by 40 states

35. 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.

36. 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.

37. • 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?

38. 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).

39. • 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.

40. • 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.

41. 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

42. 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.”

43. 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.”

44. “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?”