# Category Archives: Math

## In the News: More on Algebra!

*Here’s something we found online that seems like a good math resource for adult education practitioners:*

Check out this article/lesson plan from Patrick Honner in the New York Times:

N Ways to Apply Algebra With The New York Times *

In this article/lesson plan, Patrick shares some real world applications of math that can be investigated using information in the New York Times (or many other newspapers), such as:

**Mathematically Modeling Mortgages****Ranking and Evaluating Colleges****Calculating Car Costs****Algebra of the Election****Olympic Algebra****Solving for Stocks**

All too often workbooks teach the algebra embedded in these examples with a “one-right-way” plug-it-into-the-formula process. The examples in this article foster a much more open ended, problem-solving approach to applying Algebra in real-world settings.

This approach fosters the development of * algebraic thinking, *not just the short-term ability to plug numbers into formulas. As we stated previously:

*Algebraic thinking involves recognizing and analyzing patterns, studying and representing relationships, making generalizations, and analyzing how things change. It is about making predictions based on patterns or relationships, making decisions, and solving real problems. It is about creating models based on phenomena that occur around us. *Donna Curry (emphasis added)

**What creative and innovative approaches have you used to teach these traditionally ‘formula-based’ algebra problems? **What other types of meaningful real-world applications have you used in teaching *algebraic thinking*?

*Part of The Learning Network: (Teaching and Learning with the New York Times)

*Have you recently found something interesting ‘in the news’? If so – let us know at *eff@utk.edu.

## In the News: Combating the Negative Effects of Math Anxiety

*Here’s something we found online that seems like an interesting idea for adult education practitioners:*

Today I ran across this from Science Daily:

Math Anxiety Causes Trouble for Students as Early as First Grade

…which brings together two articles co-written by researcher Sian L. Beilock from the University of Chicago (see references below). Beilock and his co-authors have been involved in research focused on causes and solutions for math anxiety in very young learners. Aware that many adult learners describe or exhibit math anxieties (based on my own experiences and those related by adult education practitioners), I thought this information might be applicable to what we do.

One result of the research examined in this article^{1} is a better understanding of the **relationship between working memory and math anxiety** – in even the youngest learners. In the Science Daily article, Beilock is quoted as saying:

“You can think of working memory as a kind of ‘mental scratchpad’ that allows us to ‘work’ with whatever information is temporarily flowing through consciousness. It’s especially important when we have to do a math problem and juggle numbers in our head.”

Worries about math can disrupt working memory, which students could otherwise use to succeed.”

This research goes on to posit that those with stronger working memory are likely to be more affected by math anxiety (an interesting implication for those with learning disabilities in math), but Beilock also strongly cautions that:

*“Educators should not only consider math learning in terms of concepts, procedures, math curricula and instruction but also the emotions and anxieties children may bring to the learning situation.”*

We’ve touched on the importance of addressing math fears previously in a post where Donna Curry recommended that algebra fear can be combated by, “*Activities based on real-life examples, solved with concrete tools like [play] money or other manipulatives…”* More ideas can be found in another article cited by Science Daily,^{2} in which authors Beilock and Maloney state, “… *regulation* *of the negativity associated with math situations may increase math success, even for those individuals who are chronically math anxious*.”

Two techniques for helping learners to regulate or ‘reframe’ math anxiety suggested by the authors are:

**Expressive writing:** Have students write about their worries regarding math ahead of time. This is believed to help students to, “…*download worries and minimize anxiety’s effects on working memory*.” For students with low writing ability (or very young students), “…*expressive picture drawing, rather than writing, may also help lessen the burden of math anxiety*.”

**Support an emotional shift: **Anxiety is a ‘heightened’ or aroused emotional state. Teachers can help students to shift their thinking to a more advantageous heightened emotional state like anticipation. For example, “…*when students view a math test as a challenge rather than a threat*,” their performance increases as their emotions are heightened (vs anxiety which reduces performance as it grows stronger).

**What do you think? Is this information useful to you? Please share your thoughts about math anxiety!**

**What are your experiences with students with math anxiety? ***Do you have other ways you help learners to reduce their fears?*

**Have you used either of these approaches with Adult Learners?** *I’m wondering if expressive writing could also tie into strengthening writing fluency or confidence?*

*Have you recently found something interesting ‘in the news’? If so – let us know at *eff@utk.edu.

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**References:**

^{1} Ramirez, G., Gunderson, E.A., Levine, S.C. and Beilock, S.L. (2012 in press) **Math anxiety, working memory and math achievement in early elementary school**. *Journal of Cognition and Development.* Retrieved on 9/13/12 from http://home.uchicago.edu/ramirezg/RamirezG_MathAnxietyManuscript_workingdraft.doc

(*Note, I had trouble downloading this – right clicking on the link and choosing ‘save target as’ worked for me.)*

^{2} Maloney, E. A., & Beilock, S. L. (2012). **Math anxiety: who has it, why it develops**, **and how to** **guard against it*** . Trends in Cognitive Sciences*, 16(8), 404-406. Retrieved on 9/13/12 from http://www.sciencedirect.com/science/article/pii/S1364661312001465

## In the News: The Structure and Purpose of Algebra Instruction

*Have you recently found something interesting ‘in the news’? If so – let us know at *eff@utk.edu. *Here’s something we found that seems to be creating a ‘buzz’ in the adult education community:*

Aaron Kohring and Donna Curry, EFF math content experts, note that the recent New York Times op-ed piece by Andrew Hacker “*Is Algebra Necessary?*” sparked a ***lively*** discussion over on the LINCS numeracy discussion list – they recommend that you hop over and take a look at it.

In particular, these quotes from the discussion stood out:

*“Teaching math without algebra is like teaching science without the scientific method. [i.e.] ‘This is the way it is. Memorize it. No, I can’t explain how I know. This is what’s in the book, so it’s right.’ It might get you through the test, but it’s a long way from helping you understand how to use these powerful tools for your own purposes.”*

Rachel Baron GED/ABE Instructor http://lincs.ed.gov/pipermail/numeracy/2012/001333.html

“*I haven’t always been the mathlete that I am today, but math taught me to never give up, to try different approaches, to think differently about the problem I was working, and jubilation of finding a solution. Math has not depleted my brainpower but has given me more tools in which I am able to think more efficiently and effectively.” *

Brooke Istas, Moderator, LINCS Math and Numeracy List http://lincs.ed.gov/pipermail/numeracy/2012/001328.html

*“The study of algebra promotes forethought and planning, devising a systematic and logical process to derive a solution. Does this not apply in all aspects of life? I tell students that “algebra” is a planning system. When problems become more complex and require the execution of multiple steps in the correct order, we think it through and create the “plan” of execution which is the algebra equation. It is like a recipe.” *

Maureen Carro, Academic Learning Solutions http://lincs.ed.gov/pipermail/numeracy/2012/001348.html

Now some might say that subscribers to a Math and Numeracy discussion list might be biased in favor of Algebra. Here’s a post Duren found on the blogosphere from a young bibliophile and professed ‘math hater’ — also in rebuttal of the NYT article: The Fear of Math.

In the EFF online mini-course *Algebrafying Arithmetic: Developing algebraic reasoning with ALL learners* Donna Curry addresses this issue of ‘algebra fear’ and the need to re-think algebra instruction:

*“Rather than refer to ‘algebra’, we might want to talk about ‘algebraic thinking’ or ‘ algebraic reasoning’ so that we understand that it is more than just about manipulating symbols. When we understand what algebraic thinking includes, we can more readily recognize how it is used in life. Algebraic thinking involves recognizing and analyzing patterns, studying and representing relationships, making generalizations, and analyzing how things change. It is about making predictions based on patterns or relationships, making decisions, and solving real problems. It is about creating models based on phenomena that occur around us.*

*If we want our students to be proficient in algebraic reasoning, then we may need to rethink how we approach math.*

*Algebra is often taught as an abstract set of rules for manipulating cryptic symbols – and many adults have learned to fear algebra because of it. Activities based on real-life examples, solved with concrete tools like [play] money or other manipulatives, can combat these fears.**“*

Donna also notes that the National Council of Teachers of Mathematics (NCTM) advocates for algebraic reasoning to be taught and learned **at the very earliest ages** (see graphic at right).^{1} This recommendation is echoed in all levels of the EFF *Use Math to Solve Problems and Communicate Performance Continuum* for adult learners.

For more on this topic, we recommend the following NCTM resource on Algebraic thinking — cited in both the LINCS discussion and the mini-course: *A Journey in Algebraic Thinking*, by Cathy L. Seeley

**S****O – based on the original article and the various viewpoints expressed, what are YOUR thoughts on the importance of algebra?** *Is algebra useful to everyone in their daily lives or only for those continuing into postsecondary education?*

**Why teach algebra to our adult learners? ***What are some examples of algebra – or algebraic thinking – in YOUR daily life? Your students lives?*

**Comment and add your voice to the buzz!**

To learn more about teaching Algebraic thinking in the EFF online mini-course, *Algebrafying Arithmetic: Developing algebraic reasoning with ALL learners* contact us via eff@utk.edu.

**Post Contributor:**

Donna Curry, EFF Trainer & Content Expert, Center for Literacy Studies

**References and Resources**

^{1} National Council of Teachers of Mathematics, *Executive Summary: Principles and Standards for School Mathematics*, pp 3-4., Reston, VA http://www.nctm.org/uploadedFiles/Math_Standards/12752_exec_pssm.pdf

## Benchmark Numbers: Everyone Can ‘DO’ Fractions, Decimals and Percents!

**How do your students generally feel about fractions? decimals? percentages? ***Do they moan and groan and say “I hate these” or “I can’t DO… (fractions, decimals, percents)”*

**How about yourself? How much do you enjoy computations and problem solving involving fractions, decimals and percentages?**

A common challenge for many of our learners is their ability to work with a particular set of number concepts: **fractions**, **decimals**, and **percents**. Our learners have often developed ‘mental blocks’ to these mathematical concepts and even develop anxiety upon hearing the terms. Yet anyone entering an adult education classroom has already had *years *of experience in solving problems and mental math involving fractions, decimals, and percentages. *Honest!*

**Think back over the past 24 hours. How often did you use fractions? Think about a percentage? Interact with a decimal?***(other than as part of instruction)*

**Ok now, how often did you use or encounter the concept of ½? 10%? .25? ***How about quarters, dimes or dollars?*

Clearly, adults encounter and use these types of ** benchmark numbers** every day in various facets of their lives – in their work, with their families, and out in the community. See if these examples sound familiar:

*“Split that with your sister – each of you can have HALF.”**“I want to see a 100% team effort!”**“Thank you for shopping with us, you saved $3.75.”*

Helping students to realize that they *ALREADY SUCCESSSFULLY USE* fractions , decimals and percentages is one way to combat anxiety and “I can’t” attitudes. Another recommendation is to incorporate activities using these **friendly ‘fractional numbers’ **into instruction with ALL learners – even those still learning their basic math facts.^{1} Note that these recommendations are not limited to adult education – the *Common Core State Standards* *for **Mathematics* call for fractional concepts to be explicitly taught at the 3^{rd} grade level, and introduced less directly even earlier (via telling time, comparing measurements, dividing shapes into parts, etc.). These standards also state:

*“**Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.” (**CCSS-M Introduction**)*

Thus of course, as always, it is important to **keep it REAL.** Instruction should be based in everyday contexts that are meaningful to your specific group of learners. Here are some example contextual activities from the *EFF Preparing for Work Curriculum *that address fractional concepts: *A Typical Day/Time on Task*

It is also important to note that many adult learners working at more advanced levels of math instruction may have an incomplete understanding of fractional concepts. They may have memorized a process, or algorithm, but cannot easily or readily apply it to real-world situations, or easily convert from fractions to decimals to percentages. Again, some work with basic benchmarks can help – even those who think they ‘know’ fractions, etc..

** Could your students easily move from ‘80 out of 100’ to ‘80 percent’ to ‘8 tenths’?***Do they seem to confuse fractions, decimals and percentages or give up when asked?*

In *Algebraic Thinking in Adult Education *(2010), Lynda Ginsburg emphasizes the importance of ** making connections **among

**of the same information – symbols, tables, graphs, etc.**

*multiple representations*^{2}This idea applies to number concepts for fractions, decimals and percents as well. Learners need instruction and practice in understanding the equivalencies between fractions, decimals, and percentages to deepen their conceptual understanding of these numbers. Activities that mix together fractions, decimals and percentages, and/or ask students to move from one representation to its equivalent (10% to 1/10 to .1) are effective tools for both assessing and strengthening understanding.

*Comparing Numbers*is one example of such an activity (also from the

*EFF Preparing for Work Curriculum).*

**Please share with us your tips and tricks for helping adult students to understand number concepts relating to ‘fractional parts’ **(fractions, decimals or percentages)*.
*

**OR**

Try out one of the ideas in this post, and let us know how it worked for your learners(and you). Below is one last resource to help get you started!

Try out one of the ideas in this post, and let us know how it worked for your learners

*Using Benchmarks: Fractions, Decimals, and Percents – STUDENT BOOK – Lesson 5: One-tenth*^{3} http://empower.terc.edu/pdf/Using_Benchmarks.pdf

**We look forward to hearing from you and your class!**

To learn more about teaching Benchmarks in the EFF online mini-course, *How Close is Close Enough?: Improving Estimation Skills *or in the EFF *Preparing for Work *curriculum, contact us via eff@utk.edu.

**Post Contributor:**

Duren Thompson, EFFTIPS Content and Technical Editor, Center for Literacy Studies

* ***References and Resources**

^{1} National Council of Teachers of Mathematics, *Principles and Standards for School Mathematics*, (2000) Reston, VA. Pages 33-35

^{2 }Lynda Ginsburg (2010);* Algebraic Thinking in Adult Education* National Institute for Literacy, Washington, DC. http://lincs.ed.gov/publications/pdf/algebra_paper_2010V.pdf (*reference error corrected 7-27-12*)

^{3}* Using Benchmarks: Fractions, Decimals, and Percents *Schmitt, Steinback, Donovan, Merson, & Kliman (2006) Key Curriculum Press, Emeryville, CA. http://empower.terc.edu/ (Part of the EMPower mathematics Curriculum developed at TERC)