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Another great number line aligned fraction bars!

http://www.classroomfreebiestoo.com/2012/08/fractions-on-number-line.html

http://scimathmn.org/stemtc/frameworks/333a-time

http://mathcoachscorner.blogspot.in/2013/02/strategies-for-elapsed-time.html

Number lines! Elapsed time

You know I love number lines! I have also found that number lines can be a very effective way to teach elapsed time, or measuring time intervals. This method is also very helpful for word problems involving addition and subtraction of time intervals in minutes and hours.

For example: "Your trip began at 9:50 am and ended at 3:10 pm. How long were you traveling?"

When I teach students with MLD about elapsed time and numbers, I use a Think Aloud. I tell them that I start the number line with my starting time and end it with my ending time. I then look for friendly benchmarks on the number line (like jumping hours, half hours, or 15 minutes). Then, I add up all of the jumps to get my total elapsed time. Keep in mind, there are many ways to solve problems using a number line. Just make sure the student can explain why his/her answer makes sense.

Minnesota STEM Teacher Center. 3.3.3A Time. Retrieved June 6, 2013 from http://scimathmn.org/stemtc/frameworks/333a-time

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

#math learning disabilities #dyscalculia #number lines #elapsed time #math strategies

http://haileymathwiki.wikispaces.com/Frayer+Model

http://math-on-the-move.wikispaces.com/ELL+Math

#Frayer model #graphic organizer #math learning disabilities #dyscalculia

Graphic organizers.... in math??!!

Graphic organizers are not just great tools for reading comprehension! A graphic organizer is a “spatial arrangement of words (or word groups) intended to represent the conceptual organization of text” (Stull & Mayer, 2007, p. 810). The graphic organizers used in math may look different from the ones used in reading and language classes because they are sometimes used to help students visualize and organize in order to solve problems (Cavanaugh, 2011).

One of my favorite graphic organizers for math vocabulary is the Frayer Model (Frayer, Fredrick, and Klausmeier 1969; Graves, 1986). I use a combination of text and visual representations when teaching (or re-teaching) vocabulary with the Frayer Model. I especially love the "non-example" section. This helps students identify the common misconceptions (see previous post #5) and explain why it does not make sense.

Cavanaugh, M. (2011). Differentiation for Middle School and High School Math. Workshop conducted by the State Education Resource Center, Middletown, CT on October 4, 2011.

Frayer, D.A., Fredrick, W.C., & Klausmeier, H.J. (1969). A schema for testing the level of concept mastery (Working Paper No. 16). Madison: Wisconsin Research and Development Center for Cognitive Learning.

Graves, M.F. (1986). Vocabulary Learning and Instruction. Review of Research in Education 13: 49 – 89.

Stull, A.T., & Mayer, R.E. (2007). Learning by Doing Versus Learning by Viewing: Three experimental comparisons of learner-generated versus author-provided graphic organizers. Journal of Educational Psychology 99 (4): 808–820.

#math learning disabilities #dyscalculia #teaching strategies #graphic organizer

Math drills

Many teachers – and parents – hold tightly to their belief that drill is the answer for fact mastery in math (I almost got into a fight with a general education teacher at one of my training workshops over the concept of drill). However, most drill is inefficient and often negatively impacts students with MLD. To understand math, students need number sense and strategy development. Believe me, I’m all for practice in math! However, drilling without strengthening strategy use will never result in automaticity of basic facts for students with MLD.

In fact, students that have not mastered basic facts by the fifth standard are in need of something other than drill. If they have tried drill for the past 5 years and still lack automaticity, then we need to recognize our drill approach is not working!

How can we help students with fact mastery?

1. Recognize that this approach may not be effective for all students. Actually, we may be contributing to students’ negative attitudes regarding math by pushing more drill.

2. Do an inventory! Chances are... students do know some facts. For this exercise, do not let them solve the facts by counting their fingers. If they hesitate they must circle the fact that they don’t know. Assure them that you will teach them a strategy to help them to remember this fact so that they will have it memorized in the future. (Many ready-made fact inventories are available for free online. Just make sure you stick to one operation and arrange the facts in random order.)

3. Figure out the strategies that the student is already using to solve facts. How do they solve a problem they don’t have memorized? Interview the student to find out.

4. Teach a research-based strategy to help the student with basic facts – and review a few facts daily! (And, by the way, I don’t really use the term math “fact” anymore. I refer to them as “number combinations” because I work with students to see the relationships between operations. I’ll have to save that for another post.)

5. Students with severe memory deficits should use a calculator. However, students that use calculators must still self-monitor and be able to explain that their answer makes sense (more on calculator use later).

Van de Walle, J.A. & Lovin, L.H. (2006). Teaching Student-Centered Mathematics: Grades 3 – 5. Boston: Pearson Education, Inc.

#math learning disabilities #math strategies #dyscalculia #drill #math facts

#5: Using common misconceptions to anticipate mistakes and teach through non-examples

Students around the world make similar mistakes when learning new math concepts (some of this is due to our teaching mistakes and focus on procedures instead of concepts). As you are teaching, it is imperative that you keep in mind the potential misconceptions that students might have about a topic. If you are a veteran teacher, you might have these ideas stored in your memory. However, I would encourage you to begin saving examples of students’ work which exemplify these misconceptions. If you are somewhat new to teaching, there are some helpful resources on the web which will help you to anticipate misconceptions. I have found the Minnesota STEM Teacher Center (http://scimathmn.org/stemtc/) to be a great resource. Just click on the grade/standard level you teach under Mathematics Frameworks. Then, click on one of the topics in red. Go down the page, and you will find tabs, such as “Overview,” “Misconceptions,” “Resources,” etc. This site lists common misconceptions for each topic for all levels (K-12)!

As we teach or review a concept, we definitely want to use worked examples, or examples of problems that have been solved. However, we should encourage students to explain why the problem has been solved correctly. An even more powerful instructional technique is to teach concepts through a combination of worked examples and non-examples. This is where common misconceptions come in to the picture! We must incorporate common misconceptions into well-designed non-examples. When students explain why the non-examples are wrong, they are less likely to hold onto these misconceptions.

When students can explain both worked examples and non-examples, their conceptual and procedural knowledge is improved.

See the following resources for the research base behind these instructional strategies:

Booth, J.L. (2011). Why Can’t Students Get the Concept of Math? Perspectives on Language and Literacy 37 (2): 31 - 35. http://www.resourceroom.net/math/perspectives_dyslexia_math.pdf

Siegler, R. S. (2002). Microgenetic studies of self-explanations. In N. Granott & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning (pp. 31–58). New York: Cambridge University.

#math learning disabilities #dyscalculia #math strategies #misconceptions

More about word problems

As students with MLD think about the situation in a word problem, it helps for them to “see” the situation (Tucker, Singleton, & Weaver, 2006). Students must be able to visualize the problem, or represent the written information as a mental structure or idea that holds mathematical meaning. Once the student has a mental idea, he/she can move onto planning how to solve the problem and executing the necessary procedures (Foegen, 2008). However, students should not simply be instructed to “make a drawing.” They must have diagrammatic instruction so they successful translate the word problem into a meaningful representation, which shows how the parts of the problem are related (van Garderen, 2006).

One such approach to pictorially representing math word problems is Singapore Math Model Drawing. I use this with my students, along with the self-monitoring checklist, and they have shown tremendous improvement in gathering mathematical meaning from word problems. In Singapore Math Model Drawing, students are instructed to use rectangular “bars” to represent the relationship between known and unknown numerical quantities and to solve problems related to these quantities. All of a sudden, abstract words are turned into easy-to-understand pictorial models! Word problems become demystified!

The model drawing approach involves seven steps:

1. Read the entire problem.

2. Decide who/what is involved in the problem.

3. Draw unit bar (or bars of equal length).

4. Chunk the problem, placing information on model as indicated.

5. Put the question mark in place.

6. Work computations to the side or underneath.

7. Answer the question in a complete sentence.

(Roswell Independent School District Singapore Math, 2008).

With practice, you - and your students - will begin to recognize the various problem types and the associated diagrams. Happy model drawing!

Visit these websites for more information and visuals:

http://www.thinkingblocks.com/

http://my.homecampus.com.sg/

http://www.greatsource.com/singaporemath/pdf/MIF_Bar_Modeling_Position_Paper.pdf

http://www.risd.k12.nm.us/instruction/singaporemathbook.cfm

http://www.youtube.com/user/HomeCampus?annotation_id=annotation_407476&feature=iv&src_vid=qsQ-W0xZffs

Foegen, A. (2008). Algebra Progress Monitoring and Interventions for Students with Learning Disabilities. Learning Disability Quarterly 31 (4): 65 – 78.

Roswell Independent School District Singapore Math. (2008). 7 Steps Model Drawing Approach. Retrieved March 6, 2013 from http://www.risd.k12.nm.us/7%20steps%20model%20drawing%20aproach.pdf

Tucker, B.F., Singleton, A.H., & Weaver, T.L. (2006). Teaching Mathematics to ALL Children: Designing and adapting instruction to meet the needs of diverse learners (2nd ed.). Upper Saddle River, NJ: Pearson Education, Inc.

van Garderen, D. (2006). Teaching Visual Representation for Mathematics Problem Solving. In M. Montague & A.K. Jitendra, (Eds.). Teaching Mathematics to Middle School Students with Learning Difficulties, (pp. 72 – 88). New York, NY: The Guildford Press.

#math learning disabilities #dyscalculia #word problems #Singapore Math #representation

#4: Self-monitoring and word problems

One process that students with Math Learning Disabilities (MLD) typically struggle with is assessing the reasonableness of their answer (Poissant & Hirsch, 2004). Teachers can model this by estimating what their answer should be before actually solving it (Foegen, 2006). Van de Walle et al. (2013) refers to this as “thinking about the answer before solving the problem” (p. 165). For instance, students with MLD may have considerable difficulty solving word problems due to deficits in language, memory, and reasoning (Bley & Thorton, 1995; Montague, 2006). Students with MLD should think about the situation that is described in the word problem and what is happening to the quantities. In other words, students can convert a word problem into a situation (Tucker, Singleton, & Weaver, 2006). Similarly, self-monitoring is an effective strategy for helping students with MLD conquer word problems. Here are the steps (and questions to ask yourself) while working on a word problem and monitoring yourself:

READ: Are there words I don’t know? Are there number words?

RESTATE: What information is important? What is the question asking? What are the facts?

PLAN: How can I organize the facts using a diagram/model? What operation will I use? Circle or touch the sign. What might the answer be (What is the estimated answer)?

COMPUTE: What steps do I use? What is the answer? Did I get the same answer using the calculator?

CHECK: Does this answer make sense with the information I used? Is my answer close to my estimated answer? Is my answer reasonable? Did I use the correct units? (adapted from Bryant et al., 2006; Miller, Strawser, & Mercer, 1996; Montague, 2006; van Garderen, 2006).

Woodward (2006) suggests laminating the self-monitoring guiding questions rather than make students memorize the strategic prompts. I use this checklist with my students every day! Self-monitoring is sometimes included in the process of self-regulation, which includes strategies to tell yourself what to do, ask yourself questions as you solve, and check yourself (Montague, 2006).

More information about word problems to follow!

Bley, N.S., & Thorton, C.A. (1995). Teaching Mathematics to Students with Learning Disabilities (3rd ed.).Austin, TX: PRO-ED.

Bryant, D.P., Kim, S.A., Hartman, P., & Bryant, B.R. (2006). Standards-Based Mathematics Instruction and Teaching Middle School Students with Mathematical Disabilities. In M. Montague & A.K. Jitendra, (Eds.). Teaching Mathematics to Middle School Students with Learning Difficulties, (pp. 7 – 28). New York, NY: The Guildford Press.

Foegen, A. (2006). Evaluating Instructional Effectiveness: Tools and strategies for monitoring student progress. In M. Montague & A.K. Jitendra, (Eds.). Teaching Mathematics to Middle School Students with Learning Difficulties, (pp. 108 – 132). New York, NY: The Guildford Press.

Miller, S.P., Strawser, S., & Mercer, C.D. (1996). Promoting strategic math performance among students with learning disabilities. LD Forum, 21(2), 34-40. Retrieved February 21, 2013 from http://www.cldinternational.org/pdf/initiatives/mathseries/miller1.pdf.

Montague, M. (2006). Self-regulation Strategies for Better Math Performance in Middle School. In M.Montague & A.K. Jitendra, (Eds.). Teaching Mathematics to Middle School Students with Learning Difficulties, (pp. 89 – 107). New York, NY: The Guildford Press.

Poissant, H., & Hirsch, J. (2004). Using Mediated Teaching and Learning to Support Algebra Students with Learning Disabilities. Journal of Cognitive Education and Psychology 4 (1): 134-143.

Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and Middle School Mathematics:Teaching developmentally (8th ed.). Boston: Pearson Education, Inc.

van Garderen, D. (2006). Teaching Visual Representation for Mathematics Problem Solving. In M.Montague & A.K. Jitendra, (Eds.). Teaching Mathematics to Middle School Students with Learning Difficulties, (pp. 72 – 88). New York, NY: The Guildford Press.

Woodward, J. (2006). Making Reformed-Based Mathematics Work for Academically Low-Achieving Middle School Students. In M. Montague & A.K. Jitendra, (Eds.). Teaching Mathematics to Middle School Students with Learning Difficulties, (pp. 29 – 50). New York, NY: The Guildford Press.

#math learning disabilities #self-monitoring #dyscalculia #estimation

http://images.tutorvista.com/cms/images/38/decimal-places.jpg

Yikes! This is what we want to avoid - an example of showing symmetry around the decimal point. The decimal point just signifies the word "and", and it doesn't have numerical value. It does help us determine the place values of the numbers around it, but the focus should be on the ones place.

http://www.math-aids.com/Place_Value/

A place value chart to help you with decimal language. However, this chart does not show symmetry around the ones place.

Why are there no "oneths"?

As I have been working with my students to use fraction and decimal language, two of them asked me, "Mindy, if there are tenths, hundredths, and thousandths, why are there no oneths?" (One of these students was in the 5th standard, the other was in the 9th.) What a great question!

This is a very important point, especially for conceptual understanding of fractions and decimals. The best explanation I have found is from (Tucker et al., 2006).

When we teach fractions and decimals, we have to stress that there is symmetry in the decimal numeral system. The system is symmetric around the ones, or units place (see image below, page 244). The common misconception is that it is symmetric around the decimal point (Tucker et al., 2006). However, the decimal point is just a symbol to alert you to say the word "and." (Remember the last post about the connection between a decimal and a mixed number). The decimal point just helps you to identify place value, which is all centered around the unit (or ones place). Tens and tenths are just multiples and parts of the unit. I can't have a "oneths" place because the symmetry of the decimal numeral system is in base-ten, with one whole/unit as my starting point. I have to take multiples and parts that are either 10, 100, 1,000, and so on. I'm still working on the explanation.... but this is what I have so far. I hope the attached pictures help make this more clear!

#fractions #Decimals #math learning disabilities

The connection between fractions and decimals (Tucker et al., 2006, p. 246).

#3: Fraction and decimal language

This is probably my greatest pet peeve with Indian schools: the way that students are taught to "read" fractions and decimals. Students are taught to read fractions such as 3/4 as "three upon four." Instead, students should read fractions in a meaningful way, describing the numerator and denominator as parts of a whole. "Three-fourths" describes the size of the parts in the whole (fourths) and the amount of those parts (three) that we actually have.

Reading fractions in this way becomes especially important when we teach decimals. If students already read 4/10 as "four-tenths," then it is a lot easier to understand decimals meaningfully. Instead of reading 0.4 as "zero point four," students should read it as "four-tenths," which correctly identifies the place value. "Zero point four" holds no conceptual meaning.

If we focus on rote reading techniques, such as "three upon four" or "zero point four," we might think students are making quick gains. But, re-teaching will be necessary later on. It is hard to remember things are are meaningless (Tucker et al., 2006). Only if we teach students about meaningful connections between fractions and decimals (and percents) we will see them retaining knowledge.

For example, when we read 24.96 as "24 and 96 hundredths," we are actually reading a mixed number (24 96/100) (Tucker et al., 2006).

The foundational skills of fraction and decimal language will impact students' understanding of these concepts for years to come. Students who understand what fractions and decimals actually represent will be able to complete operations with fractions and decimals more accurately (because they will be able to self-monitor and explain why their answer does or does not make sense: #4 on the list).

Preliminary data analysis from my dissertation shows a HUGE gap between students' understanding and retention of fraction and decimal operations for students with math learning disabilities versus typically-achieving peers. Only 50% of students in 7th - 10th standards with math learning disabilities in my sample (N = 64) were able to subtract fractions with like denominators (4/7 - 2/7 = ?), while 97% of students (N = 62) without MLD were able to correctly subtract. When adding fractions with unlike denominators (1/2 + 1/3 = ?), only 22% of students with MLD could solve the problem accurately, while 92% of typically-achieving students obtained the correct answer of 5/6. When adding decimals, only 37% of students with MLD solved the problem correctly (8.6 + 5.42 = ?), while 92% of typically-achieving students correctly answered 14.02.

Clearly, students with math learning disabilities need different techniques to learn fractions and decimals in order to retain procedural understanding of concepts by the time they reach the end of secondary school. For more information, read my dissertation (once it is completed) and stay tuned to this blog for more instructional strategies!

#math learning disabilities #fractions #decimals

#2: Estimation

An estimate is an approximate answer that is close to the actual answer.

In order to estimate, we need to know benchmark numbers. Benchmarks are points of reference from which other measurements can be estimated. Sometimes, I describe benchmarks to my students as "landmarks" on the number line. They are places that everyone knows and has memorized on a number line. For example, on the number line between 0 and 100, the easiest benchmark to find is the half-way point, 50. Benchmarks are "friendly" numbers (Van de Walle et al., 2013) that students are comfortable with.

Knowing benchmark fractions (and decimals and percentages) are extremely helpful for students with MLD. Students should know whether the fraction (between 0 and 1) is closer to 0, 1/2, or 1. We need to use number lines to show estimation through visual representation.

Students, especially those with MLD, must estimate their answer before solving the problem. This is one of the most important steps in self-monitoring, or assessing the reasonableness of their answer (#4 on the list of more important things to teach kids in Math). Van de Walle et al. (2013) refers to this as “thinking about the answer before solving the problem” (p. 165).

For example, consider the problem: 6/7 + 5/8 =

If students use their estimation and benchmarks BEFORE solving the problem, they might reason that 6/7 is close to 1 and 5/8 is close to 1/2, so the answer should be close to 1 1/2. (Fazio & Siegler, 2011). Using estimation helps students know whether or not their final answer makes sense and can help with the accuracy of their calculations. They have a warning sign that something has gone off track, if their final answer is not close to their estimated answer.

It is not enough to just teach students how to estimate, or round numbers to the nearest whole, ten, hundred, etc. We have to teach students with MLD how to apply estimation to each and every type of math problem they encounter.

Fazio, L. & Siegler, R. (2011). Teaching Fractions. International Academy of Education. Retrieved April 11, 2013 from http://unesdoc.unesco.org/images/0021/002127/212781e.pdf.

Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and Middle School Mathematics: Teaching developmentally (8th ed.). Boston: Pearson Education, Inc.

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