Category Archives: Math
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 email@example.com.
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 article1 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 firstname.lastname@example.org.
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
Have you recently found something interesting ‘in the news’? If so – let us know at email@example.com. 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
SO – 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 firstname.lastname@example.org.
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
SO – since ‘launching’ EFFTIPS to the world earlier this week, we’ve gotten a dozen new subscribers (and a LOT of page views)!
Welcome, welcome to all!
One of the the purposes for this blog is to “…make connections among adult education practitioners implementing standards-based instruction and/or quality instructional principles. Well, this launch is certainly helping us connect!
Among our new subscribers is Kate Nonesuch, from Victoria, Canada. Kate has recently started a blog, Working in Adult Literacy, where she states:
“I’ve been working in adult literacy and numeracy for more than twenty-five years..my goal is to share everything I know about teaching before I retire.”
In reading her recent posts, we were struck by one, Not a Fairy Tale. In it she describes an incredibly patient process of waiting for a student to become comfortable enough in the adult education setting to finally overcome her fears and participate. After 3 months, the student began to both write and seek written dialogue on her writing with peers. While Kate states that there was no ‘fairy tale ending’ (because the student then moved away), Kate’s program may have, in truth, opened HUGE doors for this student, simply by patiently and politely persisting.
This story, in turn, connected with one from our recent pilot of the EFF online mini-course Line ‘Em up: Linear Functions – Graphs and Equations for ALL Learners. In a recent discussion post inside the online course, Donna Parrish, who teaches at Rogue Community College in Oregon, told this story:
In teaching an algebra focused class this summer, I have been using EMPower (Seeking Patterns, Identifying Rules, Thinking Algebraically). It has been a good learning experience for both the students and the teacher. There was one student who struggled with the ‘complete the table, describe the pattern in words, write an equation activity’ – I was afraid she would drop the class. She hung in there and when we did the ‘people seated at the banquet table’ activity* she worked and worked but never got the relationship or the equation written (only one out of the 15 students did). We spent several minutes talking about how students approached the problem and talked about just playing with the numbers, i.e., taking a stab at a statement or equation and testing it out.
The discussion was rich with comments like, “It probably doesn’t involve exponents because the number of people seated doesn’t grow fast enough.” We graphed the data and noticed that for every table, two people were added…so we could “forget” the specific points and go up 2, over 1 (slope intro!). Somewhere in the middle of all the thinking and talking, the girl who had struggled raised her hand and asked me to come to her. She had the correct equation – it was all fitting together for her! We celebrated and I danced. At the end of class she said that figuring that equation out had made her day, in fact, it was probably the best day she had ever had in school. Talk about having days made…wow!
One of the things that stood out to us in both these stories was patience. Adult educators need lots of patience – to allow their learners to ‘feel safe,’ to help them overcome years of anxiety and mental blocks, and, well, for many other reasons. For many of our learners there aren’t ‘quick fixes’ — connections come slowly. The other thing that stood out was the power of success. Making that leap — connecting with others and/or a concept — can be an AMAZING event in adult learners’ lives.
What about you? Do you have a success story where patience was key? Where ‘finally getting it’ was immensely powerful for an adult learner? If so, please share!
Post a comment and help us make connections and build this network of outstanding educators!
*This activity lays the groundwork for thinking about functions and linear equations by asking students to discuss how many people can sit at different arrangements of banquet tables. They are asked to seek a pattern as 2, 3, 4 etc. square tables are pushed together to form a row. Students are first asked to talk it through, then to build an in-out table, and then finally an equation or rule that can be used to find how many people a given number of tables can seat.
Duren Thompson, EFFTIPS Technical Editor, Center for Literacy Studies
Donna Parrish, ABS/GED Instructor, Rogue Community College in Grants Pass, OR
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 3rd 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 multiple representations of the same information – symbols, tables, graphs, etc.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).
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!
Using Benchmarks: Fractions, Decimals, and Percents – STUDENT BOOK – Lesson 5: One-tenth3 http://empower.terc.edu/pdf/Using_Benchmarks.pdf
We look forward to hearing from you and your class!
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)