Scott Adamson, Ph.D. Chandler-Gilbert Community College [email protected]






Brain Research Overview Recent Experiences Proportional Reasoning from Arithmetic Review to Diff. Eq.






Be prepared, at the end of this segment, to share one thing that impacted you. If something catches your attention, jot down a note to remind you of this. You can be impacted by
Something that surprises you
Something you disagree with
Something you have experiences with
Anything you find worthy of sharing

All possess a fundamental number sense, which he calls “numerosity.”
This inherent ability is even more basic to human nature than language is.
Numbers do not exist inside our heads the way words do: they are a separate kind of intelligence with their own brain module
Module is located in the left parietal lobe – where “math happens.”


The ability to think mathematically arose out of the same symbol-manipulating ability that was crucial to the first emergence of true language.
We should be able to “do math” as well as we speak…but we often don’t recognize when we are using mathematical reasoning.














Bring students’ ideas about how to solve or analyze problems into the public forum of the classroom. Allow students to decide whether something was mathematically reasonable. Focus on mathematical meaning – e.g. meaning of multiplication, meaning of derivative. Discuss strategies for solving “word problems.”




Teachers must engage students’ preconceptions. ▪ Math is about learning to compute. ▪ Math is about following rules that guarantee right answers. ▪ Some people can do math and some people can’t

Understanding requires factual knowledge and conceptual frameworks.
Build conceptual understanding, procedural fluency, and connected knowledge.
Build resourceful, self-regulating problem solvers.


The person in the classroom doing the most talking about the content is growing the most dendrites (brain cells).
A large number of teachers grow dendrites daily since they are the only ones talking about math!
Whether you are instructing the whole class or having students work in small groups, learning is facilitated when instructors ask open ended questions and acknowledge and encourage a variety of ideas as students engage in interactive discourse.













The significance of the dendrites in the neocortex is that their neural networks are the basis of human intelligence. According to research, these dendrites increase with use and decrease with disuse. To understand what this means in the larger scheme of things, we must employ the common cliché: "use it or lose it.“ Because there is no limit to the human's capacities to learn more, neurons are continuing to make new connections on a day to day basis throughout our lifetime. If we are learning passively, the dendrites won't create enough neural networks for enhanced learning. Active learning (or the continual use of the brain in experimentation and dialogues) on the other hand, is essential because our brains will create new neural networks that override pre-existing ones to help us make more connections, become more intelligent, and process more information.

www.newhorizons.org/

Just as phonemic awareness is a prerequisite to learning phonics and becoming a successful reader, developing number sense is a prerequisite for succeeding in mathematics.
Children in the primary grades encounter a sudden shift from their intuitive understanding of numerical quantities and counting strategies to the rote learning of arithmetic facts. Unfortunately, most children lose their intuition about arithmetic in the process











Working memory has capacity limits and time limits that teachers should keep in mind when planning lessons – less is more and shorter is better! Information is most likely to get stored if it makes sense and has meaning. If teachers cannot answer the question, “Why do we need to know this?” in a way that is meaningful to students, then we need to rethink why we are teaching that item at all!

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument.
Mathematics is about problems, and problems must be made the focus of a student’s mathematical life. Painfully and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process – having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work.


The truly painful thing about the way mathematics is taught in school is not just what is missing – the fact that there is no actual math being done in our math classes – but what is there in its place: the confused heap of destructive disinformation known as the “mathematics curriculum.”
“The area of a triangle is equal to one-half its base times its height.” Students are asked to memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill, the joy, even the pain and frustration of the creative act. The question has been asked and answered at the same time – there is nothing left for the student to do.


Electrical Brain Stimulation May Boost Math Skills




Adults did better on tests after painless therapy, but more research needed, experts say


Arizona Republic November 7, 2010

Those of you who are lousy at math may someday be able to boost your skills with the use of a painless method of electrical brain stimulation, British research suggests. Adults with normal math abilities were able to improve their performance on a series of numerical tests with the help of a noninvasive technique known as transcranial direct current stimulation (TDCS).






Pair up and share one thing that impacted you. Explain why/how this thing impacted you. Each person has 2 minutes to share.

1 1 1 ÷ 2 4

Dan Meyer - http://www.youtube.com/watch?v=BlvKWEvKSi8

Facts and Skills Facts and Skills

Facts and Skills

Facts and Skills

What does it mean to multiply?
3x2
¼x6
¾x½

1 1 1 ÷ 2 4




What is a fraction?
Part to whole…OK but that’s not all!
Imagine…cutting and iterating…multiplicative




What does it mean to divide?
How many copies?

1 1 1 ÷ 2 4

x−2 = m+n 2 x −1 = m + n `

x = m + n +1 `

x−2 = m+n 2

x−2 = m+n 2 x − 2 = 2(m + n ) `

x = 2(m + n ) + 2 `

A square measures 4x2y3 on a side. The area of a square is A=(side)2. What is the area of the square?
An attempt to provide a “real-world” example so students can practice a skill.
The bottom line? Blah, blah, blah…find (4x2y3)2…just because.





What are the meanings that are lacking?
Do we know the meanings or are we good at the

procedures?




Do we have the time to help the student to develop these meanings?
There is so much material to cover…
Students are ill prepared…




What if the student doesn’t care?


The student will not really develop a well connected

network of understanding if they are not willing to build upon meaning, purpose, making sense, reasoning, thinking, and conceptual understanding!

Lamon, S. J. (2005). Teaching fractions and ratios for understanding. New York: Rutledge

Which tree, A or B, grew more? Explain your reasoning. Lamon, S. J. (2005). Teaching fractions and ratios for understanding. New York: Rutledge

Focus on the 2006 situation. How much taller is Tree B than Tree A? Lamon, S. J. (2005). Teaching fractions and ratios for understanding. New York: Routledge

Sue and Julie were running around a track equally fast. Sue started before Julie. When Sue had run 9 laps, Julie had run 3 laps. How far had Sue run when Julie had run 15 laps? Cramer, K. & Post, T. (1993, February). Making connections: A Case for Proportionality. Arithmetic Teacher, 60(6), 342-346.

Sue and Julie were running around a track equally fast. Sue started before Julie. When Sue had run 9 laps, Julie had run 3 laps. How far had Sue run when Julie had run 15 laps?

9 x = 3 15 3 x = 9 ⋅15 3 x = 135 x = 45

09540K

ADDITIVE RELATIONSHIP





The number of laps that Sue runs is always 6 greater than the number of laps that Julie runs. The runners are running at the same speed.

MULTIPLICATIVE RELATIONSHIP





The number of laps that Sue runs is always 3 times as large as the number of laps that Julie runs. The runners are not running at the same speed!

Sue and Julie were running around a track equally fast. Sue started before Julie. When Sue had run 9 laps, Julie had run 3 laps. How far had Sue run when Julie had run 15 laps?

9 x = 3 15 3 x = 9 ⋅15 3 x = 135 x = 45

Suppose you pay $8,000 per year for health insurance. You are informed that your health insurance costs will drop by 3000%. What does this mean?

http://www.youtube.com/watch?v=lUd-slJc-GY

You have subscribed to Verizon broadband internet access. Your contract calls for a rate of $0.002 per kilobyte of use while in Canada (that is, the data transfer rate is $0.002 per kilobyte). How much will you pay for 35896 kilobytes of use? Explain how our meaning of multiplication plays a role in this problem.

http://www.youtube.com/watch?v=zN9LZ3ojnxY


Can you see 3/5 of

something in this picture? Be specific.
Can you see 5/3 of something in this picture? Be specific.
Can you see 2/3 of something?
Can you see 1÷3/5? Thompson, P. (1995). Notation, Convention, and Quantity in Elementary Mathematics







If you ask a 3rd grade student what 3/5 means and if they could draw a picture of it, what do you think their response would be? If you ask a 3rd grade student what 5/3 means and if they could draw a picture of it, what do you think their response would be?










A typical response is that 3/5 means “3 out of 5 things,” but not three-fifths (three one-fifths) or 3 pieces, each 1/5 of a whole. The typical response is an additive response, not multiplicative. A typical response if they have not worked with “improper” fractions yet is that 5/3 doesn’t make sense, because you can’t have 5 out of 3 things. Other students just show and explain 3/5 again. The point is that fractions should be taught the same way regardless of the size of the numerator or denominator. The same is true for the four basic operations. Students shouldn’t have to think differently just because of the size or type of numbers involved.

In a research study, 12-15 year old students were asked how to find the cost of .22 gallons of gasoline if one gallon costs $1.20. (They were only asked to indicate the operation and not to perform the calculation). ▪ The most common answer was 1.20 / .22. ▪ When the same question was asked with "easy" numbers, e.g., find the cost of 5 gallons of gasoline if one gallon costs $2.00, the students answered correctly: 2 x 5. Taken from Andrew T. Wilson - Austin Peay State University

What are the misconceptions?


Students develop misconceptions about multiplicative reasoning.
The cost of .22 gallons must be less than the cost of one gallon, therefore divide, since division makes smaller.
Multiplication makes larger.
Divisor must be smaller than dividend.

Taken from Andrew T. Wilson - Austin Peay State University

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What should students be practicing and how shall they practice? If we want students to reason proportionally, we should ask them to reason proportionally! Students should practice exactly what we want them to be able to do. If we want them to reason, then we should have them practice reasoning. If we want them to practice a procedure to get right answers, then we should do that! According to research about how the brain behave, the person in the classroom who is doing the most talking about the content is growing the most dendrites (brain cells). Who is growing dendrites in your classroom? Research shows that understanding develops during the process of solving problems in which important math concepts and skills are embedded.

http://www.youtube.com/watch?v=4xpTjMhdIA0









Allow children to work out their own ways to solve problems involving multiplicative thinking. Compare additive and multiplicative thinking approaches Use models that clearly illustrate the idea/s Make and discuss the links between fraction ideas, rates, ratios and proportion Use authentic contexts and models to exemplify situations

Taken from Jeanne Carroll – Victoria University, Australia








Be clear about the relationships between the ideas. This is not straight-forward, the more I think about the ideas the more insight I have about them Use common sense as well as number sense Estimation is really important as it demonstrates understanding of the concepts involved

Taken from Jeanne Carroll – Victoria University, Australia







Engage in conversations about the ideas and talk about the links, discuss where are the similarities and differences between the ideas It is development of fuller, deeper and more connected understandings of the number system that makes a difference

Taken from Jeanne Carroll – Victoria University, Australia

Part I

You are traveling in your car down the freeway with the cruise control set. You observe your odometer reading to be 8300 miles and that your car has consumed 4 gallons of gasoline. Your car gets 28 miles per gallon. Create a linear function that models the odometer reading as a function of gasoline consumed. Write a detailed explanation of what this linear function represents in the context of the situation.

In the previous problem situation, the idea of constant rate of change comes up (28 miles per gallon). What do we mean, generally, by constant rate of change? To say that an object moves at a constant speed The meaning of proportional means that any distance it travels is in correspondence is that if an object moves proportional correspondence with the amount of some distance in some amount of time, time in which it travels that distance. then in any fraction of that time it moves the same fraction of that distance, and vice Put another way, if the object travels D distanceversa. units in T time-units at a constant speed, then in a/b*T time-units it moves a/b*D distance-units.

Part II In Part I, you were asked to define a function so that its graph passes through the point(4, 8300) with a rate of change of 28. This figure shows one function that does this. A) B) C) D)

What does 28 stand for in this figure? What does –4 stand for in this figure? What does (–4)28 stand for in this figure? What does (–4)28+8300 stand for in this figure? E) What does 28x + (–4)28+8300 stand for in this figure?

2. Do the following for each of a-e. Check your answers on your graphing calculator. Define a function in the form y = ax + b so that its graph passes through a) the point (5,2) with a rate of change of 7. b) the point (–3, 4) with a rate of change of –2. c) the point (2.73,–5.15) with a rate of change of 7.26. d) the point (–4.1,-6.8) with a rate of change of 8.6 e) the point (–2,–1) with a rate of change of -4.2

y = 7.26.x + (-5.15) – 7.26(2.73)) y = 7.26.x – 24.97

3. Generalize your previous work by writing a function definition for line that passes through the point (j,k) with a constant rate of change of h. Create a graphical representation that could be used to help explain this general function definition.

i units

-h*i units

y = h.x + j – hi y – j = h . (x – i)

For some painkillers, the size of the dose, D, given depends on the weight of the patient, W. Thus, D = f (W), where D is in milligrams and W is in pounds.
Interpret the statements f (140) = 120 and f ′(140) = 3 in terms of this painkiller.
Use the information in the statements above to estimate f (145). Hughes-Hallett, Calculus: Single and Multivariable, 5/e

f (145) ≈ 120 + 3(5)