Scale Diagrams as a Testing Strategy and a Teaching Opportunity

by Melissa Braaten

The end of the calendar year is the season for HSE testing, so I’ve had a lot of students looking for test prep recently.  Every practice test seems to include at least one Pythagorean Theorem question with an accompanying diagram, like the one below:

Sarah has to drive from her home to the post office, then on to the grocery store before she returns home.  She will travel on the roads shown below.  How many miles does she drive altogether?

Picture1

It did not surprise me that my students mostly didn’t recognize that this problem could be solved with the Pythagorean Theorem, since I had not yet taught them this.  I was surprised, however, at how few students attempted to estimate the missing side in order to come up with a reasonable distance for Sarah’s trip.

While not all diagrams on tests and in textbooks are drawn to scale, many are, and this can be a valuable reasoning tool.  (Often, diagrams that are not to scale are labeled “not to scale.”) I decided to try to assess students’ propensity and ability to estimate a missing side on a scale diagram. I gave 16 adult basic education students (of all reading and math levels) the problem below:

Picture2

 

Estimate the length of the side with a ?, and explain how you came up with your estimate.

 

 

Of the 16 students given this question:Picture3

  • 5 gave no answer at all, either leaving it blank or writing “I don’t know.”
  • 2 gave answers that were not reasonable and appeared to have been the result of a calculation (one student wrote “I added both sides.”
  • 4 wrote reasonable amounts, then added “I don’t know” or “Not sure” or in one case, erased the number.
  • 5 wrote reasonable amounts and gave either no explanation or a reasonable one.

I offered all of my students one hour of instruction on estimating with scale diagrams. I gave them sticky notes and showed them how to use the length of the labeled sides (marked on the sticky) to estimate the length of the unlabeled sides. Three weeks later, I gave them the same trapezoid question again.

Of the 14 students who answered the post-assessment question:

  • 2 out of the 14 (14%) did worse.
  • 7 out of the 14 (50%) showed improvement: 3 went from no answer to a reasonable one, 2 went from unreasonable answers to reasonable ones, and 2 went from reasonable answers they doubted to reasonable answers without doubting.

The results of this informal data suggest that a relatively minor intervention (1 hour of instruction) led to improvement for a significant number of students in their ability or willingness to use proportional reasoning to estimate the missing sides on scale diagrams. This could help them both with test taking as well as geometry instruction, since the ability to estimate reasonable lengths can help students make sense of geometric shapes, quantities, and formulas.

The relatively high incidence of students who were able to come up with a reasonable estimate, but felt the need to qualify that they didn’t know or weren’t sure, suggested to me that many students didn’t so much lack the ability to estimate proportionally, but were not aware that this was something they were “allowed” to do in math. A few of the students who got unreasonable estimates did so by improperly adding or multiplying, probably remembering that these are things they were often asked to do (for area and perimeter, perhaps) and did not recognize or were not bothered by the fact that the resulting numbers made no sense.

As teachers, we should let our students know that it is appropriate and encouraged to use spatial and proportional reasoning to estimate with scale diagrams. This will improve their comprehension of geometry concepts, push their proportional reasoning, possibly lead to better test taking, and hopefully, above all, reinforce the idea that mathematics is about making sense of our world.

————————————————————————————————————————-

===================================================================

Melissa BraatenMelissa Braaten is an adult education instructor at Catholic Charities Haitian Multi-Services Center in Dorchester, MA. Melissa has taught ASE and pre-ASE math and reading, as well as ABE writing, computer skills, and health classes. Melissa also is a training and curriculum development specialist for the SABES PD Center for Mathematics and Adult Numeracy at TERC.

Advertisements

Differentiating Instruction with Parallel Tasks

by Sarah Lonberg-Lew & Melissa Braaten

One of the biggest challenges we face teaching math in adult education classes is having students who vary widely in their readiness, prior knowledge, and reasoning ability. Ask a question of your whole class and some students will have their hands up with the answer before others have made sense of the question. The situation often feels intractable – how can you keep all your students in that sweet spot of being challenged without being frustrated, and feeling confident without being bored? The two-word answer is: differentiate instruction – teach the same material to multiple levels at the same time. However, it’s easy to say, but hard to do.

Luckily, the math concepts that we teach in our classes can and should be taught at multiple levels. Fractions and data are not one-time topics, but rather ideas that should be returned to and deepened as students move through the levels. Geometry, measurement, and proportional reasoning are threads that run through the whole curriculum. Algebraic reasoning can begin to develop at the lowest levels and continue to grow. But how do we address these topics at multiple levels, all at the same time, and  in the same classroom?

One useful strategy to differentiate instruction is to use parallel tasks. These are pairs of questions or tasks that address the same big idea, but at different levels of difficulty. Students can choose between the tasks or even do both if they are so inspired. Struggling students can build confidence by experiencing success with a more accessible task and faster-moving students can be challenged with a more sophisticated application. After completing the tasks, the class can come back together to discuss the overall big idea that was at the heart of both tasks. In addition to being an activity where students are neither bored nor overwhelmed, parallel tasks also contribute to student confidence by giving them a degree of agency in their education. Choices are empowering.

One way to create parallel tasks is to change the numbers involved in the problem. The chart below shows some characteristics that make numbers more accessible or more challenging to work with:


Blog34 image


 

Here is an example of a pair of parallel tasks that uses this approach:

Task 1 (More accessible):

To make homemade toothpaste, Tanya mixes 2 teaspoons of water with 8 teaspoons of baking soda.

Juan mixes 4 teaspoons of water with 10 teaspoons of baking soda.

Will both mixtures have the same consistency?  Why or why not?

Task 2 (More challenging):

The pool regulations call for 11 oz. of chlorine to be added daily for every 10,000 gallons of water in the pool.

The East Bush Pool contains 25,000 gallons of water. The pool technician added 33 oz. of chlorine for the day. Did she add too much, too little, or just the right amount of chlorine to the pool? How do you know?

Notice that in this example, both tasks get at the big idea of determining whether two ratios are equivalent, even though the contexts are not the same. It’s okay to use different contexts as long as all students are working on the same concept.

Another factor that affects the difficulty level of a problem is the number of steps it takes to solve. Problems which require many different steps will also require more planning, organization, and strategic thinking on the part of the student and will usually be more challenging. Yet another factor is familiarity — a problem can be more or less challenging for students depending on whether or not it relates to their personal experiences or knowledge.

Below is an example of parallel tasks that have different levels of complexity. Both tasks address the concept of comparing unit rates. While both tasks involve calculating and comparing unit rates, the second one requires students to make decisions about unit conversions before calculating unit rates.

Task 1 (More accessible):

Marianne ran 100 meters in 20 seconds.

Tony ran 200 meters in 50 seconds.

Who is faster?  How much faster?

Task 2 (More challenging):

Marianne biked 50 miles in 3 hours and 20 minutes.

Tony biked 75 miles in 4 hours and 10 minutes.

Who is faster?  How much faster?

There are no hard and fast “rules” about how to use parallel tasks.  As one teacher reflected on her experience using parallel tasks in her classroom:

“I offer both options and let students choose. Sometimes I say which is more challenging, sometimes I don’t. If someone does the easier one quickly, I ask them to try the other as well. If someone tries the harder one and gets stuck, I suggest they do the other one first. Since people are self-differentiating, I don’t usually have any issues with self-esteem or buy in. I find it works quite well, especially for warm ups.”  

Some valuable resources for teachers:

Marian Small has a series of books called Good Questions: Great Ways to Differentiate Mathematics Instruction which contain both open questions and parallel tasks as well as advice for implementing them in the classroom.

Good Questions for Math Teaching, Grades 5-8: Why Ask Them and What to Ask by Lainie Schuster and Nancy Canavan Anderson also contains many great ideas for open questions and others that could be used as parallel tasks.

====================================================

sarahlonberg-lew

Sarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. She teaches in an adult education program in Gloucester, MA. Sarah’s work with the SABES PD Center for Mathematics and Adult Numeracy at TERC includes developing and facilitating trainings and assisting programs with curriculum development. She is the treasurer for the Adult Numeracy Network.

Melissa Braaten

Melissa Braaten is an adult education instructor in the Greater Boston area. Melissa has taught ASE and pre-ASE math and reading, as well as ABE writing, computer skills, and health classes. Melissa also is a training and curriculum development specialist for the SABES PD Center for Mathematics and Adult Numeracy at TERC.

Why is Reading Math So Hard? (aka, My attempts to assess why my students struggle to read word problems)

by Melissa Braaten

It’s common knowledge that many adult students struggle with word problems – which, incidentally, make up the majority of the questions they will be asked to answer on high stakes HSE exams.  Since word problems bring together both language and mathematical reasoning, they require students to use and integrate several skill sets.  Deficits in any of these skills can cause students to get lost. A lot of literature on word problems involves helping students build operation sense (the ability to know what the operations can look like in the real world in order to select the correct ones to use), building mathematical vocabulary (as distinct from a focus on “key words,” which can be misleading), and problem solving strategies.

While the skills mentioned above are indispensable and probably account for a great deal of the difficulty that students encounter with word problems, I have also found, in many students, an additional difficulty that seems to be distinct.  I have worked with students who appear to have the necessary mathematical and vocabulary foundations to approach a word problem, and who have demonstrated problem solving acumen in other contexts, and yet they are still completely lost reading a word problem.  It appeared as if, although they could decode the words and even know what the words meant, they still couldn’t understand what they were reading.  This led me to wonder: Is reading math questions different than reading other types of text?  I suspected it was, but wanted to learn more.

To attempt to assess “math reading ability” in isolation, I took HSE style word problems and wrote three options for paraphrasing the question from the word problem.  Two of the options were not a correct paraphrase, and one was.  I asked students not to solve the word problem, but only to identify which of the choices was asking the same thing as the original question.  Students struggled quite bit with these exercises.

Some of the “easier” examples could be identified by matching a basic unit:

Question 1:
St. Thomas’ School has decided to put tile in the math classroom.  The classroom is 12 feet x 15 feet.  The tiles come in boxes, and each box will cover 6 square feet of floor.  How many boxes are needed?

A. How many tiles come in a box?

B. What is the area of the classroom that will be covered in tile?

C. How many boxes of tiles will it take to cover the area of the classroom?

In the question above, students could have chosen the correct answer merely by identifying that option c is the only question that asks for a number of boxes.  When I tested this question with 16 adult education students with varying levels of math and reading ability, 13/16 or 81% chose the correct answer.

Questions that involved more complex units like rate were harder.

Question 2:

Folders come in packs of 10.  St. Thomas’ needs 4 folders per student, and expects to enroll 20 students in September. Folders cost $11 for one pack.  How much will St. Thomas spend for folders per student?

A. How many folders will St. Thomas need to buy for September?

B. What is the cost per folder?

C. What is the cost to buy 4 folders for one student?

When I put this question through a readability checker to test for vocabulary and sentence complexity, it was given a GLE of 3.2.[1]  Every student I gave this to has a reading comprehension of at least GLE 4 (and some up to 11), yet this question was only answered correctly by 6 students, or 38% of the group.  Of those who answered incorrectly, 7 chose option a, a question which asks for a number of folders, which doesn’t match the unit of the original question.

What is going on here?  It would take more detailed and careful research to answer that question.  To identify that option c was asking the same question as the original, students would have to realize that “How much did St. Thomas spend” and “What is the cost” are asking for the same type of unit, and they would also have to equate “per student” with “for one student.”

I wanted to see if explicit instruction in identifying the unit in a question and defining the word “per” would help students with this type of task.  The same group of students received one hour of instruction and practice in identifying the units in a question and identifying that per described a unit rate.  Four weeks later, they were given question 2 again.  In the post-test of 14 of the original 16 students, four of the students who had originally answered incorrectly now chose the right answer, while one person who got it right the first time got it wrong.  Another way to see it is that 8 out of the 14 post-testers got the question right after instruction, or 57%, which is a modest improvement.

While my informal classroom “research” needs a lot more work to tells us anything definitive about what skills students may be missing and how to intervene, it does suggest that the ability to read math problems is distinct from overall reading ability and that instructional interventions might be helpful.  I hope to encourage more interest in this question so we can find ways to help students overcome this barrier.  If you have your own observations or interventions, or have encountered useful research in this area, please share below in the comments!

————————————————————————————————————————-

[1] https://readable.io/text/

===================================================================

Melissa BraatenMelissa Braaten is an adult education instructor at St. Mary’s Center for Women and Children in Dorchester, MA. Melissa has taught ASE and pre-ASE math and reading, as well as ABE writing, computer skills, and health classes. Melissa also is a training and curriculum development specialist for the SABES PD Center for Mathematics and Adult Numeracy at TERC.

Having Realistic Expectations

by Sarah Lonberg-Lew

As a math teacher in adult education classrooms, I have never had the experience of having a class of students who had similar educational backgrounds and ability levels. In fact, I don’t think I’ve ever met a math teacher of adult learners who has had that experience. And yet, when I plan my curriculum and lessons, I write one set of learning objectives for all my students. But how can I expect them to arrive at the same place at the end of a lesson when their starting points are so vastly different? It feels unfair to my students, as if I am asking them to run a race but giving some a head start and asking others to carry heavy weights. And those students at the back, the ones who have struggled the most to get here, who have had the fewest opportunities to learn – they are the ones who have to work the hardest to master the objectives.

Should I then lower my expectations for those students? Should I write them a different set of learning objectives, possibly sending them the message that I don’t believe they are as capable as their classmates? I think the answer is these students should get the same lesson as the others with the same learning objectives and the same high expectations regarding effort and engagement, but that it is a disservice to them and to me as their teacher to hold the expectation that they will arrive at the same level of mastery as their peers. It is not realistic, and neither I nor the students should be pinning our ideas of what it means to be successful to something that is not realistic.

I believe it is important to be honest with my students. If I pretend that all it takes is grit and effort to move from fourth grade level skills to college readiness in one semester, then if my struggling students fail to achieve that, they may believe it is because they didn’t try hard enough. The truth is that it takes children nine years to go from fourth grade to twelfth and that’s when their education is uninterrupted and when being a student is their only job. Many of our adult learners can move faster than that, but it is still a huge task and one that could take years.

I believe in my students. I want them to believe in themselves and one way I can help them do that is by making sure we all have realistic expectations. When my students get discouraged because they don’t understand something as well or don’t get an answer as fast as the person sitting next to them, I believe it is important to acknowledge that they did not start at the same place, that they have had their own unique challenges and that it doesn’t make sense, in light of that, to expect that they would arrive at the same place.

A student handed me an assignment the other day and said, “I don’t think I did a very good job.” I asked her, “Did you do your best?” She said she had. I asked again, “Did you give this your absolute best effort?” She said she had. “Then,” I said, “it has to be good enough. Nobody can do better than their best. How could I ask for more?” I told her I’d give her some feedback and help her improve – we can always make our best even better – but that she should feel proud of her effort and accomplishment.

Because I expect my students to address the same learning objectives at different levels, I strive to provide multiple entry points to the material by differentiating instruction as best I can. I try to create a classroom experience where everyone can be challenged and no one feels bored or overwhelmed. I seek out new methods and strategies to bring my classroom closer to this ideal. But differentiating instruction is hard. It sometimes feels impossible. I give it my best effort and know that in the moment I can’t do better than my best, but I can keep working to make my best even better.


sarahlonberg-lewSarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. She teaches in adult education programs in both Gloucester and Danvers, MA. Sarah’s work with the SABES numeracy team includes facilitating trainings and assisting programs with curriculum development. She is also an actively involved member of the Adult Numeracy Network.

 

 

Paying Attention to Student Work

by Melissa Braaten

Effective teaching of mathematics uses evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.  National Council of Teachers of Mathematics, 2014[1]

The quote above is a good description of “formative assessment” in mathematics, in which assessment information is sought for the purpose of informing and modifying instruction.  Paying attention to the work our students are handing in can provide some of this valuable “evidence” of student thinking.  But what does that really look like?  How much can you glean from a few things that a student writes on paper?

It can be helpful for teachers to have a structure to guide their analysis of student work. One of these methods is known as the DEN process, which stands for Describe, Evaluate, and Next Steps. The idea is to observe carefully and thoroughly (Describe), before making inferences about what the student may or may not understand (Evaluate). Lastly, the teacher plans her Next Steps with that student to untangle misunderstandings or push conceptual understanding further.

Let’s look at an example of how even a small sample of student work could provide insight and potential next steps for a teacher. The two samples below come from an adult education class at an intermediate level (GLE 5-8). Both students answered the question correctly, but looking closely at their work provides details about their understanding of operations, integers, and notation.

STUDENT A

Picture1

Describe

In this sample, the student has recorded calculations horizontally, except for the last one. The last calculation is recorded vertically, but not lined up. The student has used subtraction to record the checks and addition to record the deposit, and has also included an equal sign after each step in the calculations. All the expressions are connected by these equal signs.

Evaluate

The student uses the equal sign the way it would be used on a calculator, to indicate the “answer” or result of a calculation, rather than to indicate equality between expressions (since the connected expressions are not equal). The final calculation was most likely written that way because the student was running out of room, not because she used the vertical format to perform the calculation, seeing as the place values are not aligned and there is no evidence of borrowing or other strategies which would be necessary for that particular subtraction problem. Although the student used a negative integer (-18) to represent when the balance of the account was in debt, she was more comfortable recording the check transactions using subtraction, rather than as negative integers. The student followed the chronology of the problem very carefully, even to the point of including the two checks together in the first calculation (probably because the existence of two checks is mentioned before the amounts).

Next Steps

This student does need some instruction in the proper use of the equal sign, since an understanding of equal as equality or balance between two expressions is critical for algebraic manipulations. In addition, this student could be encouraged to think about other ways that the problem could have been solved. While calculating each step chronologically works fine for this situation, there are many scenarios and word problems in which it is advantageous to work the numbers out of the order in which they are mentioned. Using negative integers instead of subtraction could help with this, since the addition of positive and negative integers can be done in any order.

STUDENT B

Student B

Describe

The student added the initial amount and the deposit, then (in two separate steps) added the three check amounts. Lastly, the student subtracted the check amount from the total of the initial amount plus deposit. Each calculation was recorded as a separate equation with a single number on the right.  The check amounts were represented as positive.

Evaluate

Student B appears to be more flexible than Student A in approaching the problem. Instead of calculating chronologically, she subtotaled the credits and the debits to the account. As a bonus, she never had to calculate a negative account balance or use negative integers at all. The notation is both horizontal and correct, but having the calculations broken up the way they are may indicate that the student is uncomfortable with writing multistep equations.

Next Steps

This student could work on combining the many steps of calculations into one multistep equation.  Teaching the student to use parentheses to group amounts in an equation would be helpful for representing the problem the way she solved it.

Although both students seemed to understand the word problem and answered it correctly, a closer examination still revealed areas in which a teacher might want to probe further.  Hopefully these examples illustrate the kind of rich formative assessment than can be gleaned from a very small sample of (correct) answers!

————————————————————————————————————————-

[1] Principles to Actions: Ensuring Mathematical Success for All. NCTM. 2014. p 53.
=============================================================

Melissa BraatenMelissa Braaten is an adult education instructor at St. Mary’s Center for Women and Children in Dorchester, MA. Melissa has taught ASE and pre-ASE math and reading, as well as ABE writing, computer skills, and health classes. Melissa also is a training and curriculum development specialist for the SABES PD Center for Mathematics and Adult Numeracy at TERC.

Connecting Math to ESOL

by Ann Pellagrini

I’ve always had a love/hate relationship with math.  In school, even though I was in advanced math classes, I struggled to grasp the concepts because teachers didn’t connect what we were learning to real-world applications. To this day, the only things I remember about geometry are the Escher drawings in the book. Consequently, if there’s someone in the room who seems to know more or has some kind of credential like being a math teacher or engineer, I tend to defer to them for the answers because I don’t always trust that I’ll have the correct answer (even though I often do). So it’s no surprise that I’m not a math teacher. However, I am an ESOL adult education instructor teaching students with educational levels ranging from third grade to graduate school but who nevertheless are in the same class together.

The necessity to include math in my ESOL classes became apparent when my Advanced class had to refer to the information in a simple chart in their book to answer questions. Some of my most educated students were giving answers that were not contained in the chart, nor could be found anywhere else on the page.

My idea to connect math to ESOL began with an activity in which students would create charts as a way for them to collect and analyze the data. After pre-teaching topic-related vocabulary, I had the students collect specific information about each other. Modeling as we went, I had them answer a series of questions about their classmates that they would find in their charts such as which language is spoken the most/least, from which countries are the most/least of the students, etc.  Once we got through this activity, I decided to try and expand it to do some percentage work.  That I found to be much more challenging and so I made a mental note to introduce and incorporate percentages in my teaching throughout future lessons as it applied.

One way I did this was in a monthly review of students’ class attendance in both my Intermediate and Advanced classes.  I wanted students to be more aware of their attendance and how it relates to and affects their educational progress.  So, at the beginning of every month, students were given a monthly calendar on which they would track their attendance. At the end of the month, we calculated the total number of classes and they calculated the total number of classes they attended.  Utilizing the symbols we had reviewed in earlier lessons, together we created the formula for calculating their individual attendance percentages which they listed on their calendars.  Half way through the school year, I gave them a line graph of our monthly classes and gave a lesson on how to plot their actual attendance over those months.  Not only did they “get” why they had to track their monthly attendance, but they were able to “see” how their attendance related to the number of classes given to date.  Additionally, by analyzing the attendance data through discussion, the class was able to think about how different events such as illness, work, and weather affected their attendance and whether they needed to do anything different or get some additional help to ensure them having strong attendance through the remainder of the year.

This exercise gave them a real-world application of something they were doing (attending class) as well as a sense of ownership and control of their class attendance. (See sample student chart below.)

30_attendance sample chart

I also found another way to incorporate math into my Intermediate class. When I stumbled upon the fact that students did not know their height or weight in U.S. measurements, it became an opportunity to introduce them to rulers, yardsticks, and measuring tapes. My quilt-maker was teaching those in her group how to use a soft tape measure while my construction guys (who often have low formal education levels) were teaching their classmates how to use a retractable metal one.

Once they had their heights, students added them to the large chart I made for the classroom and we talked about what they saw. They compared their heights to that of their partners, compared the heights of the two gender groups, determined which of the men and women were of similar heights, answered whether 5’6” is greater or less than 5’3” and whether 1 cm is > (greater than) or < (less than) 1 in.

I had students come to the front of the class to be put in order of height by their classmates or to compare back to back who was taller, shorter, or were the same heights.  They loved this activity and they were now able to know their height in both metric and U.S. measurements – and they did some work on superlative adjectives, too!

NOTE: If you do this activity, do not use the men to illustrate shortness because it may embarrass them and shut down the activity. Also, if you have students from cultures where men and women can’t be in close proximity, only do same gender activities.

In both these classes, making things fun AND having real-world application prevented any fear of doing math because the students were focused on finding the answers to questions that were useful and/or interesting to them while enjoying working together with their teams. Real-world application, especially when tied to the students’ experiences, makes for impactful learning.

I recently was asked by a math teacher how I handled situations when students are not engaged in the learning, and I had no answer for him because my students are engaged and interested. Don’t get me wrong, there are times in lessons when this is not the case and that’s when I know I need to find a way to engage them and/or do something more interactive. However, with the math that I’ve introduced to my students, they don’t have the opportunity to not be engaged because it’s very interactive. When students voice concern that they’ve never seen this or don’t understand that, I ask them to be my helpers for modeling the activity or they get paired with someone who has stronger skills. Along the way, everyone gets reminded that they can do this because they are all smart and we are all learning together.

When you set the bar high and tell students that they can reach it, they often believe it and will work toward that goal.

In the past seven years of teaching, I am amazed by what is expected of teachers in adult education. In no other industry that I’ve worked have I seen so many demands on put on the workforce with so little or no compensations or benefits. For this reason, I want to offer a simple an easy way for teachers to connect math to ESOL. While interacting with students, listen and look for those places were a little bit of math could make for some interesting discussions. Is there and opportunity to compare the percent of men and women who like [insert item or activity here]? What about a pie chart when discussing food or showing ½ an apple?

Bring in some math and let the students have fun finding answers to relatable situations. You’ll be surprised that you are connecting math to ESOL and the students won’t even know it. Shhh… let’s not tell them!


Pellagrini photo

Ann Pellagrini began teaching at age 9 in her parent’s sunporch.  After 20+ years working in various management levels in government, high tech and retail, she returned to Education as an adult ESOL teacher where she sneaks in math as often as possible.

What’s the Big Idea?

by Connie Rivera

You may have seen one version or another of Did You Know?, a video that went viral in 2007. I’ve seen it more than once during presentations since WIOA regulations came out. The key take away for me was the idea presented in the video that “We are currently preparing students for jobs that don’t yet exist, using technologies that haven’t been invented, in order to solve problems we don’t even know are problems yet.”

How can we teach our students in such a way that they will be prepared for something we don’t even know about yet?  I think that one of the most important things we can do is base our instruction on big mathematical ideas – making connections to core content rather than getting too caught up in answers to specific problems. Other disciplines use the phrase enduring understandings, which I think highlights that an understanding which is core and true today lasts into the future.

When I was first a math student, math was full of disconnected topics that I needed to memorize. I was decent at that, so I just made it through. However, we now know from research classics like The Teaching Gap and Adding It Up: Helping Children Learn Mathematics that effective instruction connects understandings across different math content and across levels. This quote validates why teaching differently, with big ideas always in sight, matters.  As Hiebert and Carpenter have said, “We understand something if we see how it is related or connected to other things we know[1]… The degree of understanding is determined by the number and strength of connections”[2]. Let’s link instruction and understanding to big, central ideas for our students!

You may be thinking OK, tell me what those big ideas are and I’ll teach them! But that’s the thing – there is no single answer to what the big ideas are, and big ideas are bigger than the size of a lesson, or even a unit.  Still, we can plan for them when we are planning our instruction. One way I’ve collected lists of big ideas is through conversation with other teachers about what those big ideas are and how understandings connect . I like to use sticky notes for this brainstorm, with one sticky for each idea, so we can organize and re-organize our ideas. At first we come up with topics like basic operations, fractions, and equations. Then we group them and think about what’s bigger and crosses into other content. Ideas connected to equivalence, place value, properties of operations, proportional reasoning, and algebraic thinking come to the surface. These are the things that are important for our students to understand. Maybe you can even have your students make connections and develop understanding by doing this same activity! (See links to articles for other lists.)

Understanding big ideas is motivating and builds ways to transfer knowledge to new problems our students will need to be prepared to tackle. To read more about big ideas, check out “Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics” and “What is Mathematical Beauty:  Teaching Through Big Ideas and Connections”. If you want to experience teaching for understanding that is connected to the big ideas, consider taking Building a Solid Foundation or one of our other offerings.


[1] Hiebert, J., & Carpenter, T. P. (1997). Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann.
[2] Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Mcmillan.

==========================================================

connie riveraConnie Rivera teaches numeracy skills to adults of various skill levels, including court-involved youth and English Language Learners. She is also a math consultant, providing math strategies and support to programs implementing the College and Career Readiness  Standards for Adult Education (CCSRAE). As a consultant for the SABES numeracy team, Connie facilitates trainings and guides teachers in curriculum development. Connie is President of the Adult Numeracy Network and a LINCS national trainer for math and numeracy.

The Misbegotten Trifle: What Cooking Fails Have in Common with Math Learning

by Sarah Lonberg-Lew

In a Thanksgiving episode of the popular nineties sitcom Friends, the character Rachel attempts to make a traditional English trifle. The layers include: ladyfingers, jam, custard, raspberries, beef sautéed with peas and onions, bananas, and whipped cream. As she lists them, the other characters’ reactions go from appreciative to incredulous to disgusted, and the mention of the beef layer gets a big laugh from the studio audience. Upon investigating, her friends discover that the pages of the magazine from which she got the recipe were stuck together and she had in fact made half a trifle and half a shepherd’s pie.

The scenario gets such a big laugh because most of us know it is absurd to have beef sautéed with peas and onions in the middle of a dessert, but Rachel clearly did not. If she had, she might have questioned the recipe or even dared to modify it on her own. As it is, it turns out she did find it a bit odd, but she had more faith in the recipe than in her own intuitions. (Click here to see the “trifle scene.” Warning: The clip contains some PG dialog.)

Sadly, this is exactly the situation many of our students are in. Lacking conceptual understanding or a desire to seek it, they rely almost exclusively on procedures, believing that faithfully executing steps, practicing and memorizing them, is the only way to reach their math goals. They don’t even pause when they achieve an absurd result because they have no concept of what kinds of answers make sense.

The reasons they believe so strongly in steps are not insignificant. For one thing, many have never known that there can be any more to math than following steps. For another, it really feels like it’s working in the short term. By teaching students steps to memorize and then giving them practice problems that require nothing more than applying them, both the teacher and the students can feel successful in the moment. However, when it comes to long term retention, knowing when an answer is reasonable, or the ability to solve non-routine problems or apply their learning outside of class, the time and energy invested in memorizing steps fails to serve our students in any useful way. In fact it is counter-productive because they have missed the chance to use their time to develop genuine understanding and they end up having to “learn” the same material again and again. (See “What Community College Students Understand about Mathematics” by Stigler et. al. for an in depth explanation of research supporting this.)

I recently had an opportunity to observe a student struggling through finding half of 544. He knew the steps and diligently used long division to divide by 2, even going through the procedure twice to be sure. Both times he arrived at an answer of 322 (having made the same mistake twice) and said he felt confident about the answer because he had followed the steps (he really said that!). We then talked a little about what half means and I asked him if he could find a way to check whether 322 was really half of 544. He multiplied it by 2 and was completely stymied when it did not come out to 544. His steps had failed him and he had nowhere else to go.

Instead of reviewing the steps or combing through them for his mistake, I lent some context to the question, asking how he would manage if we had $544 to share equally between us. Immediately his energy and attitude changed – I could see in his face that he understood the task and had an idea of how to approach it. He began talking about what kinds of bills the money could be in, and I kept pace with his thoughts by drawing pictures of the bills as he talked. When he saw them in front of him, he drew loops to divide them into two groups, even deciding to go to the bank to make change when the bills couldn’t be evenly divided. When I asked him this time what reason he had to feel confident in his answer, he pointed to the two groups, showing me that they were both the same and that he had “given out” all the money. Using his conceptual understanding of division, he arrived at the correct answer with confidence and was proud of his hard work. I followed up with another “finding half” problem and then asked him to see if he could use his strategy to divide a number into three equal parts which he did with relative ease. By this time, the strategy was becoming a little more abstract – just numbers in boxes instead of pictures of bills– and eventually even the boxes fell away. It was especially fascinating to watch him because his approach evolved into something that I could not have taught him. It was his own invention born of his own understanding.

It is not just our students who suffer from the misconception that math is all about steps. Most of us have also been socialized to believe that the key to success in math is being able to follow directions. I remember my tenth grade math lesson on completing the square (a strategy for solving quadratic equations). My teacher drilled us over and over again on the sequence of steps and said, “Keep your pencil moving! Don’t think!” I was overwhelmed and lost and despite many other successes in math in the years following, I believed for a long time that I would never understand completing the square. I just couldn’t remember all those steps. When it came time to teach it in my own classroom (during my stint in K-12) I did only marginally better with my students than my teacher had done with me, giving lip service to explaining the algebraic steps and trying to be more gentle and compassionate with them as they struggled to memorize them. It was even longer before I made sense of the method for myself and saw that it was in no way beyond my ability.

As teachers, we need to stay strong when our students beg us to just give them the steps – when they say, “I know if you just show me how to do it, I’ll understand.” I have had students who have insisted that memorizing steps is “how they learn best,” but even those students who are good at memorizing benefit more from understanding. For all of us there is a limit to how much we can remember – but not to how much we can understand.

We have to take the long view and know that even getting a perfect score on a worksheet is not the same as understanding. Taking the time to develop conceptual understanding can feel slow and our students are usually in a hurry to move forward with their academic and career goals. But learning math takes time just like any other kind of meaningful learning. For our students, the difference between spending their time learning deeply and spending it memorizing steps may very well be the difference between progress toward their goals and one more failure in one more ABE class.

The misbegotten trifle came about because of a blind reliance on following steps. The consequences of such an approach for our students may be far worse than an unpalatable dessert!


sarahlonberg-lewSarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. She teaches in adult education programs in both Gloucester and Danvers, MA. Sarah’s work with the SABES numeracy team includes facilitating trainings and assisting programs with curriculum development. She is also an actively involved member of the Adult Numeracy Network.

 

 

Teaching Math to ESOL Learners: A Reflection

For this installment we’re featuring a special guest blog by Sister Margaret Lanen of Notre Dame Education Center in South Boston (http://www.ndecboston.org)

When I was asked to teach a math class once a week this year, I was most willing. I taught algebra, geometry and advanced math in high school for many years and thought it would be easy to teach basic math to new English language learners.  Little did I think that the language used in math would be such a stumbling block to ESOL students! I thought it would not be very difficult to teach math vocabulary and processes. What a surprise!

The students were nervous about their lack of math skills when they began the class. I started with the basic operations of addition, subtraction, multiplication and division. It didn’t take long before I realized how many words like minus, sum, times, and divided by were new vocabulary to many of the students. Trying to solve a division problem during the first weeks (especially with some of the Haitian students, who use a different division procedure than we use in the U.S.) made me realize that teaching math to ESOL students was not going to be so easy. Students would tell me they knew the answer, but they couldn’t explain it in English. It is challenging to get the right vocabulary and language to express mathematical thinking in another language.

Luckily, our program director had agreed to invite folks from the SABES PD Center for Mathematics and Adult Numeracy to come to our site for math professional development. When I participated in the workshops, I realized how different my experience as a high school math teacher was from teaching math to adult language students. The trainer introduced a whole new way of teaching math which was hands-on. This was brand new to me.

She began with the staff working together on basic mathematical concepts. The emphasis was always on how we got the answer. Staff worked in groups placing fractions on a specially created number line. We had many activities which included using scissors, graph paper, string, magic markers, rulers, etc. We were learning by doing.

How did this affect my teaching of math? Well, now in every class I always include a group activity to give the students a hands-on experience of the skill I am teaching. It is a challenge for me to plan the class. For example, I knew that the concept of pi would be difficult to get across. With string, rulers, cans of different sizes, and scissors the students measured the circumference of the can and the length across the center. The three groups found that the ratio of the two measurements were the same. This experience gave the students a visual measure of pi as a number a little bigger than 3.

I do feel a sense of community as the students work together to help one another. I’ve learned I need to be patient and give the students time to get started with an activity and decide what they are going to do. Group work often illustrates that there are different ways of getting an answer.

One big fear of some of the students when they started the class was that they would fail math. The best part of teaching this math class is that the students do not need to pass a test to succeed. They come to see the many ways math can help in their daily life. I bring newspaper ads and they determine how to take 25% off the cost of an item at the store. We learn to read graphs such as the history of immigration in the United States. They use graph paper to decide how much fencing they need for a garden or what is the best plot size for the vegetables. Many math concepts develop from these activities.

When I hear the conversation of students in their groups helping one another to understand a math concept, I feel excited. My sense is that the students are learning math and feeling more confident and I am learning, too.

What Does It Mean for Math to Be Relevant to Adult Learners?

by Melissa Braaten

When I am trying to prioritize my curriculum and to build buy-in from my students, I am always trying to think about how to make the math relevant to them. But what does relevance mean?  Is it relevant for someone to learn math that they might use someday in a career they are considering? Are the specific algebraic procedures what they will use in the future, or the deeper algebraic reasoning and problem solving that they develop in the process? Is a particular math concept relevant if a student will not apply it in practice for years?

Some of the challenges faced by making math relevant involve time gaps between learning and application. In addition, research has shown[1] that the math used in careers is often specific to that career, and methods are “idiosyncratic” and learned on the job. In this article, I will describe different types of relevance that I aim for as an adult basic education math teacher when designing and prioritizing my curriculum, and which types of so-called “relevance” seem to bolster the argument that “school math” is something that real adults never use.

Immediate relevance to adult and family survival

When I think about immediate relevance, I am thinking about math that can be applied to contexts that adults are engaged in already, and in which they already have a felt need for more mathematical understanding. A few examples of this would be math that involves:

  • shopping and consumer awareness (math skills: understanding percentages as used in sales and taxes, proportional reasoning to determine the best deal)
  • financial literacy (math skills: basic operations for budgeting, decimals and percentages to understand banking, credit, savings, and loans)
  • home and appliance use and maintenance (math skills: reading gauges, understanding measurement systems, types, and conversions to purchase and calibrate products for the home, car, etc.)

The list could go on, and will also vary a bit from person to person depending on the contexts in which they are currently operating.

This type of relevance has (in my experience) the greatest effect on student buy-in. In addition, increased numeracy skills in these areas can help families materially improve their lives. When planning a unit, I try to include at least one application which will fall into the category of immediate relevance for most of my students.

Relevance for expanding horizons

This next category will vary quite a bit from student to student. There are many applications for math (especially the math content taught at levels GLE 1-8) than can be applied immediately by most adults to improve or enrich their lives, but which may not already be a felt need because the person has not previously been engaged in that context (possibly because of lack of math skills!). An example for many of my students would be home improvement. All kinds of simple home improvement and repairs require skills of estimation, measurement, numerical reasoning, fractions and decimals, or geometric concepts of area, perimeter, and volume. I have found that many of my adult students, who often lack some of these math skills, also avoid engaging in this context. Having gotten by so far, they may not come to me with a felt need for these skills, but I am hoping the experience may actually expand the types of activities they feel comfortable exploring or engaging in.

Here are some other examples of contexts which tend to be expansions for many of my students (they can immediately apply these skills, but may not have previously thought about doing so):

  • Banking and personal finance, such as comparing and opening accounts, building credit, saving for retirement, or filing taxes (math skills: numerical reasoning, positive and negative numbers, deep understanding of decimals and percentages, linear and non-linear growth)
  • Making choices based on risk or probabilistic reasoning, in contexts such as medical decisions, investment choices, or even understanding weather predictions (math skills: probability concepts, ratios, data and sampling)
  • Evaluating statistical and data based claims, which are often present in the news and necessary to follow policy arguments

These are just some examples. Often, the skills that tend to fall under immediate relevance are those that pertain to day to day survival, while the skills listed above may not be necessary to survive, but are certainly important for adults to fully and independently thrive and participate in society. At my current program, many of my students are legally homeless or dealing with significant economic instability, so the applications of math to expand their horizons might not feel as immediately important to them, but I want to give them the opportunity to begin to see what else is out there, and to know that they too have the ability to reason through and participate in these contexts. This is a way in which adult numeracy education can invest in long-term change in families and communities.

Timely relevance to educational next steps

Many math skills are also important for academic reasons, because they are built upon in later units or because the skills are important for a different discipline (data analysis in science, for example). Relevance to educational next steps, be it the next unit or the next level or to achieve a certain certification (high school equivalency or Accuplacer level), is an important reason to teach math concepts and should not be overlooked. Credentials matter to the job market and our students know it.

While I plan all of my units with future units, test preparedness, and interdisciplinary applications in mind, I do this with two important caveats. The first is that while this type of relevance is necessary, in adult basic education I believe it is also not sufficient to justify teaching a certain topic. There are so many topics in the College and Career Readiness Standards for Adult Education (CCRSAE) levels A-C with both immediate relevance and expanding relevance as well, that to leave out the first two does a disservice to our students. Academics for most adult students is not an end in itself but a means to a better life. Including the first two types of relevance along with preparing students for the next educational requirement allows them to start to use math to better their life now, instead of seeing the payoff always in the distant future.

The other caveat for me is that these connections must be timely. When I am planning levels and units and sequences, I would like students to apply their math skills to other academic contexts within a six-month window, whenever possible. Without timely use and retrieval, those math skills become something forgotten in one more notebook, gathering dust.

“Math I never use in real life”

When math is considered one of the most important skills for financial and career advancement in our society, how is it that so many adults (including me!) still feel that they never use much of the math they learned outside of school? How can we teach to avoid making math so “irrelevant”?

I don’t have a simple answer, but a couple of suggestions:

  1. Intentionally weaving in applications of math with applications of the first two types of relevance helps adults see that numeracy is something they can benefit from now and in the future.
  2. Advanced math with specific career applications should be taught later in the educational sequence, when students are already committed to a specific career path and are going to use the math in that career in a timelier fashion.
  3. Prioritizing the development of mathematical practices (such as the Standards for Mathematical Practice in the CCRS), number and operation sense, and proportional and algebraic reasoning over specific procedures.[2] These words describe thinking patterns, foundational concepts, and strategies for thinking that contribute to numeracy at every level and are relevant to any career context. Knowledge of specific procedures degrades over time if they are not used frequently, but can be easily relearned or figured out if solid mathematical reasoning is already in place.

It’s not easy to make decisions about what to teach when there is so much math and so little time. Since most adults do not learn math for its own sake but as a means to achieve other life and career goals, it is important that we teach them math that is both useful and relevant. In the process, we may even affect some long-term change in attitudes about what mathematics is all about.

————————————————————————————————————————-

[1] See National Institute for Literacy, (2010) Algebraic Thinking in Adult Education, Washington, DC. p 3. Available at https://lincs.ed.gov/publications/pdf/algebra_paper_2010V.pdf
[2] “Adults often say they have never used the algebra they learned in school.  That may be true for the rote aspects of manipulating symbols, but they likely are using the mathematical reasoning and problem-solving aspects of algebra unconsciously.” From National Institute for Literacy (2010), ibid.

===================================================================

Melissa BraatenMelissa Braaten is an adult education instructor at St. Mary’s Center for Women and Children in Dorchester, MA. Melissa has taught ASE and pre-ASE math and reading, as well as ABE writing, computer skills, and health classes. Melissa also is a training and curriculum development specialist for the SABES PD Center for Mathematics and Adult Numeracy at TERC.