Back in the early 1980′s, when I was working on that MS, I did a research paper on the topic and saw that classes in “Reading in the Content Area” were starting to be taught at Northern Arizona University.  What happened to this movement? The research I read at the time indicated that students might learn better if they were taught these techniques and/or if instructors adopted some reading techniques in their teaching. Over the years I have tried this and it worked.  My students tell me they understand my explanations . They make sense to them. What am I doing different than other instructors? Well, I try to introduce vocabulary at the beginning of the lesson whenever possible. If it isn’t, and often it doesn’t make sense until we encounter it in context, I stop and explain the vocabulary.  I try different methods, depending on the vocabulary including using the word in an everyday context or giving the Latin roots of words. In the beginning, I use everyday synonyms with the vocabulary words. I may use the synonym the first day and then start giving the synonym after the word for a few more days.  Finally I drop the synonym completely.

What else can I, as a math instructor do? With word problems I give them an aim for their reading.  I use the basic technique I learned in my high school geometry class. I have them write down what they are given and what they need to find. I have them put things together that go together – like the 15 liters and the 25% solution. Then I have them write down what they need to find.  Finally I have them start to ask themselves questions to help decide what action to take.   These questions start out at a basic level and gradually go “up the line” and include higher level thinking skills.  Are there any key words?  (With the key words, do you do the action indicated or is the inverse operation needed?) Is there a formula that uses both the information I have and what I need to find?  Can I translate directly into math symbols? Can I draw a picture or diagram? These actions help them read the problem and figure out what they need to do.   Most of the  problem solving ‘lists of steps’ I have seen skip the help with the reading and that is what helps them learn what to do.

Reading a math textbook is hard – even for an old, experienced instructor like myself.  Often, in an attempt to be clear and show the students one method that works, a text will give the most complicated method available. This can confuse even me.  Starting out with some easy problems to help a student get an intuitive idea of the concept first can work wonders.  When teaching reducing fractions, do you start with some simple examples such as those with money or pies or do you first introduce prime numbers and breaking composite numbers into prime factors so that you can cancel.  I introduce the easy ones, usually involving money for my adult students. I also introduce a few basic divisibility tests prior to this topic and use them in the lesson.  Finally I introduce the way we use when it’s not an easy problem to do – when breaking it down into prime numbers makes sense.

Reading in the math classroom is hard for today’s students.  They don’t have the background needed for it. Therefore we, as instructors, need to help them. What’s the best way to get through to them? We model what they should do. We show them how we do it.  If that means teaching reading in the  math classroom, then so be it.

Students will always do as we do – not as we tell them to do.

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Yesterday, a friend commented about the word problems in my Kindergarten assessment and how unbelievable it was that Kindergartners were expected to solve problems of this complexity.  When I looked back at the assessment, I realized that I had only included two word problems and they were not, in my opinion, of the most difficult type.

I was aware of different types of word problems, but it wasn’t until I read Table 1 in the Glossary of the Common Core State Standards for Mathematics (p. 88) that I understood the distinctions. (http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf, p. 88)  Not only are there four categories of word problems, (1) add to, (2) take from, (3) put together/take apart and (4) compare, but each category has different cases as well.  For example, in “add to” and “take from” problems the result can be unknown, the change can be unknown, or the start can be unknown.

Yes, my two measly word problems did not even scratch the surface.

But then I had another thought.  I had been focused on creating problems that were grade-level appropriate.  What was I expecting students to show me for a solution?

I tackled this and I came up with a drawing for each of my two word problems that is straightforward and makes sense to me.  For example here is my take from, start unknown problem:  3 children leave the party.  4 are still there.  How many were at the party before? As a solution to this problem, I drew the picture at the beginning of this post.  Not bad.  Makes sense.

The more difficult problems are the compare problems.  Here is an example of one of the more difficult problems taken from the Common Core State Standards for Mathematics: “Lucy has 3 fewer apples than Julie.  Lucy has two apples.  How many apples does Julie have?” (p. 88).  That was the fewer version of a compare problem with a bigger unknown.  Now, here is the more version:  “Julie has 3 more apples than Lucy.  Lucy has 2 apples.  How many apples does Julie have?”  See how much easier it is to understand the more version?

I tried but couldn’t figure out a simple drawing that would work for the fewer version of the compare problem.  I can solve it numerically, of course, but had trouble coming up with a good visual explanation.  So, I researched models for comparison word problems and came up with several resources that model this kind of problem with a pair of bars. Thinkingblocks.com (http://www.thinkingblocks.com/TB_AS/tb_as3.html) has a video along with sample problems.  The Minnesota Stem Teaching Center has a good discussion of several types of word problem models, and a little more than half-way down the page is a discussion of comparison problems using bar models (http://scimathmn.org/stemtc/resources/mathematics-best-practices/modeling-word-problems).

Based on the concepts from these resources, I created a drawing for the fewer type of comparison word problem that combines both pictures and bars.  It makes sense to me after-the fact, but doesn’t seem as intuitive to me as the other drawings.

Common Core State Standards for Mathematics, p. 88

I think this approach would have to be taught.  Which brings up another question:  Are Kindergartners or even first graders solving problems like this now?  It’s been several years since my daughter was in Kindergarten, but I don’t remember problems like this.  If Common Core standards in post-primary grades presume these skills, are students prepared?  If not, how are they going to catch up?

I have a feeling some of the Common Core word problems I’m creating may be a bit of a challenge for the students I work with.  As a result, that part of my assessment may be more a jumping off point for teaching how to recognize and solve different types of word problems than an assessment of what students know.  But that’s okay.  After all, my main purpose is to find out where the holes are and fill them – even if the holes are “new” ones that exist because of the emphasis on understanding in the Common Core State Standards.  To quote Martha Stewart:  “It’s a good thing.”

It tells about two teachers who frequently employ open-ended problem-solving sessions in their teaching – and the students (almost all) like it well and are very motivated.

In math education, OPEN-ENDED problem usually means it doesn’t have a specific step-by-step solution. You can solve it in many different ways. Or, it may have more than one solution.

The problems these teachers use are often from real life, and not quick to solve. Instead it can take some time and struggling to get anywhere. (Hey, that’s how problems in real life often are, too!)

But, struggling can be valuable. One of the teachers featured in the article, Heidi Ewer, says:

“Struggling helps them see this as an investment of their own time and energy. It makes them more willing to learn,” Ewer says. “Struggling to solve problems requires students to use their intuitive skills to investigate concepts, she explains,
and, in this way, they gain a deeper and more lasting understanding of the mathematics.”

There’s even some value in trying impossible problems! Douglas Twitchell says in The Value of ‘Impossible’ Problems, “…if you have students who are interested in trying, think of the mathematics they might learn in the process of attempting these! [some very difficult problems]“

I realize dealing with open-ended problems is not easy to do if you’re not an experienced math teacher – and not even then. Like the other teacher from the article, Judith Carter, says, a problem-solving activity is not something she can fit into every day, or even every week.

But,maybe an occasional session dedicated to a challenging open-ended math problem can be fitted into a scheme of work.

If you’re ready to give some challenging problems for your student(s), check out these:
Open-Ended Math Problems from The Franklin Institute Online (middle school level).

Here are some more

Problem Solving Decks 
Includes a deck of problem cards for grades 1-8, student sheets, and solutions. Many of these problems are best solved with calculators. All of these problems lend themselves to students telling and writing about their thinking.

mathlearnnc.sharpschool.com/cms/One.aspx?portalId=4507283&pageId=5856325

Math Stars Problem Solving Newsletter (grades 1-8)
These newsletters are a fantastic, printable resource for problesm solve and their solutions.
mathlearnnc.sharpschool.com/cms/One.aspx?portalId=4507283&pageId=5856332

Virtual Math Club
Problem sets & puzzles similar to those found on math contests such as the AMC 8, AMC 10, MATHCOUNTS, or the middle school math olympiads, including answers and video solutions posted a week later. For middle school/early high school level.
virtualmathclub.wordpress.com

Open-Ended Math Problems
Collection of problems that lend themselves to more than one way of solving.
fi.edu/school/math2/index.html

Mathematics Enrichment – nrich.maths.org
Open-ended, investigative math challenges for all levels.
nrich.maths.org/public

Math Circle Presentations
Math circle presentations for grades 6-12 from University of Waterloo and their related student exercises, available as PDF files. These can be used as enrichment, as challenging word problems or as review of certain topics.
cemc.math.uwaterloo.ca/events/mathcircle_presentations.html

Figure This! Math Challenges for Families
Word problems related to real life. They don’t always have all the information but you have to estimate and think. For each problem, there is a hint, other related problems, and interesting trivia. Website supported by National Council of Teachers of Mathematics.
www.figurethis.org

MathsChallenge.net
Search for number, geometry, probability etc. word problems and challenges. Includes solutions.
mathschallenge.net

Massachusetts Tests for Educator Licensure, Mathematics Subtest
This is a downloadable math practice test for prospective elementary school teachers, and it contains a lot of good problems for problem solving practice, mostly in the middle school, some in the high school level.
www.mtel.nesinc.com/PDFs/MA_FLD003_SubtestII_PRACTICE_TEST.pdf

See also:

• Word Problems Solving Strategies
  Gives one example of each strategy: Find a pattern, Make a table, Work backwards, Guess and check, Draw a picture, Make a list, Write a number sentence, Use logical reasoning.
  www.mathstories.com/strategies.htmMath Word Problems Worksheets
  Free worksheets from About.com for various grades in PDF form.
  math.about.com/od/wordproblem1/Worksheets_for_Word_Problems_Various_grades.htm

Award-winning mathematical word problems with different solution strategies. Interactive image maps on various human body systems with definitions as well as multiple-choice questions with instant feedback.
Via www.vtaide.com

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• Second Grade Worksheets – Free Printable Worksheets for 2nd Grade

Gulnar has an average score of 87 after 6 tests. What does Gulnar need to get on the next test to finish with an average of 78 on all 7 tests?

The website provides an option of getting hints. This was the first thing I looked at.

Hint 1: Since the average scores of the first 6 tests is 87, the sum of the scores of the first 6 tests is 6*87=522

So far so good. But the next hint appears to be designed to force the student back on the “known”  method of solution :

Hint 2:  If Gulnar gets a score of   x on the 7th test, then the average score on all 7 tests will be :

(522+x)/7

Followed  by a more logical and timely

Hint 3: This average needs to be equal to 78 so:

(522+x)/7 = 78

and by the answer in the next hint:

Hint 4:

x=24

And now I would like to make a few comments about  these hints.

The Gulnar problem is a classical  three steps problem of elementary school arithmetic  (more precisely, two-and-half steps problem, because the two first steps are almost identical, as I will show you soon; in teaching, it is useful to emphasise the benefits of re-use of the same step). It is a pleasure for me to revisit the art of “questioning”, the key ingredient of solving word problems as it was taught in Russian elementary schools of my time:

Question 1. How many points in total Gulnar got in 6 tests?

Answer: 6*87=522

Question 2. How many points in total  Gulnar needs to get in 7 tests?

Answer: 7*78 = 546

Question 3. How many points Gulnar needs to get in the 7th test?

Answer: 546 – 522 = 24

In the encapsulation/de-encapsulation terminology, what is needed for solving the problem (and therefore what is interiorised in the student’s mind)  is

• de-encapsulation of the concept of an  average, and
• reversing the operation of forming an average.

Mastering this two  two mental actions is needed not so much for further development of the theme which uses “an average” as a starting point, but for better understanding of the very concept of an average.  In my opinion, this key methodological point appears to be missed by the writer of hints provided on the Khan Academy website.

Please notice that in the Khan Academy’s hints, Hint 2 is in itself a multi-step problem. Most likely, it is rooted in the material which is marked as prerequisite for “Average word problems”, namely “Systems of linear equations” preceded by “Linear equations 3″ preceded by “Linear equations” 2 and 1 preceded by “Adding and subtracting fractions” and “Dividing fractions” and so on up to “Addition 1″. In my opinion, the nature of hints restricted to recently learned material suggest a strict modular structure of material. As my solution shows, a synoptic approach with back references to much earlier material (basic subtraction and multiplication) could be didactically more useful.

And my last comment: when I was a child, I was taught to start solving problems like that by asking questions (which I would describe in my adult parlance of nowadays  as meta-questions):

Question 0′.  ”Gulnar has an average score of 87 after 6 tests. ”  What questions can be asked about these data?

Question 0”.  ”Gulnar needs to get  an average of 78 on all 7 tests“.  What questions can be asked about these data?

Basically, I and my peers were taught to do logical analysis of data (and of data structure). This explains the great didactic value of word problems.

Of course, I have to add a disclaimer: it is wrong to judge a systematically developed learning material after looking at the very first randomly chosen exercise. I would not write a similar post about a random exercise in a textbook (I mean, a real textbook, printed on paper). Almost by default, textbooks are linearly structured. On the contrary, the highly modular structure of the Academy’s website makes it open to assessment of  randomly chosen nodes in its intricate dependency of topics flowchart (“Knowledge map”).