Sticker math fun (K-2)

When I was a kid, my mom sometimes bought my brother and me activity books to fill up our summer time. Well, if you're into activity books, here's a Sticker Math Fun by Usborne that I can recommend for summer math learning and practice (grades K-2) (or for later in the school year as well, for that matter).

It has different mathematically-sound activities for addition, subtraction, beginning multiplication, fractions and clock, mostly on about 1st grade level but some for kindergarten and some for 2nd.

It is filled with quite variable and creative activities, and not just dull simple adding and subtracting. The activities include missing addends or subtrahends, number patterns, finding numbers that add up to a certain number, and so on.

It is in full color with kid-friendly imagery.

We bought this book several months ago, and I can say my kindergartner-1st grader has really enjoyed the book! And the third grader has wanted to do several pages as well.

The normal price is $15, but this morning when I checked, you can get copies from Amazon marketplace sellers for less than $5! Go check it out!

Office furniture giveaway

Here is your chance to win a piece of office furniture in this little giveaway contest:

• First go check www.csnOfficeFurniture.com website and find a piece you'd like to have if you are the lucky winner. It needs to be $135 or less in value.
  
  
• Then, to participate, purchase any of the Math Mammoth or Make It Real Learning products at Kagi store between now and July 9, 2009.
  
  
• In the comments field on the order page, put either the name of the piece of furniture you'd like to win or the URL to its web page. If you leave this comment field blank, I will not know that you wish to participate, so it is crucial!
  

That's it! I will have a random draw for ONE winner within the participants, and put you in contact with csn Office Furniture company. They will pay for the shipping within United States.

Links you might need:

• www.csnOfficeFurniture.com to check what piece of furniture you'd like to win


• Math Mammoth Blue Series information and order page


• Math Mammoth Golden Series and Green Series information and order page


• Math Mammoth Light Blue Series information and order page


• Math Mammoth discounted packages / CDs information and order page


• Make It Real Learning workbooks information and order page.

Math Teachers At Play #10

Welcome to the tenth edition of Math Teachers At Play blog carnival! Here with the heat of the summer, "less is more". We concentrate on teaching issues, but also get to "play" a bit with binary numbers, geometry, integers, and an optical illusion.

TEACHING

What's a math teacher going to do with Wolfram | Alpha ?



In case you haven't heard... Wolfram|Alpha is a new, computational search engine. And if you're a math teacher, you should be aware of it.

Collection of Web Freebies discusses Wolfram|Alpha as your Personal Free Online Math Assistant.

I wrote an introduction to Wolfram|Alpha as well. I feel it can both be a benefit and a drawback to math education - a benefit because it can free us from routine calculations, and a drawback because students still need to learn to do those, but how can you easily enforce that?

W|A presents a dilemma to math teachers... because it can solve SO MUCH. It can solve just about any routine type calculation in algebra 1 or algebra 2 or calculus courses. So what can a teacher do? Jason, The Number Warrior wonders how we can be Teaching to the Limits of Wolfram Alpha. In essence, should teachers strive to find problems that cannot be solved with Wolfram|Alpha?


What's a math teacher going to do with homework?

Photo courtesy of Vicki's pics

Another, an age-old, dilemma for math teachers. What is the best way to assign, grade, and check homework? Sam Shah conducted an unofficial survey on the matter and presents the Homework Survey Results.

I glanced over the survey results, and found one response that really "struck" me as a "sweet" solution to the homework problem. It is from a 5th grade teacher, and I apologize for taking so much space for it here but I just feel it was really good (and if the person feels I shouldn't quote this, please let me know). Emphasis mine:

"I work hard at the beginning to set up a culture of responsibility for themselves. They are responsible for making sure they understand, for asking questions, for asking for more practice or another example.

When we come into class, we check the answers together. I will solve a problem or two on the board that I have learned over the years is confusing. Other than that, I expect them to ask for clarification if a problem is wrong.

They "grade" their own. If it's wrong, they are expected to take a minute right then to figure out why it was wrong. Usually it's a silly mistake. If not, they bookmark it to ask questions during our next work time.

I don't give any grades. I don't assess it. The final test/quiz is their grade. The homework is just them figuring out how to do the skills. They know they won't be graded on it. They know they won't be penalized for getting it wrong. They know that the whole point of the homework is to figure this math stuff out. If they get it wrong, they know they haven't mastered it and need to get help.

On the opposite side, if a student doesn't do their homework, I don't get too upset. We have a talk about how they hurt themselves, because now they don't have an accurate measure of if they can do it on their own. I can't help them because they didn't get the opportunity to make their mistakes before the test. However, I have a few who don't need to do homework. They still earn A's on the final test. They learned it by being in class and paying attention.

With those who obviously do need the homework practice, but aren't doing it, I problem solve that on a case-by-case basis.

It changes the whole culture of learning and mastery in the classroom. But it is VERY important to set up that culture at the beginning."

Now, I'm not claiming that middle and high school teachers could easily transfer these ideas to their classes... but perhaps they can, at least somewhat. Please discuss homework at Sam Shah's post.

A teacher's problems do not end with homework. Can you believe that TEACHING as a profession is so difficult in today's world that sites exist solely to help new teachers to SURVIVE their first year? That's exactly the word that is used. Teaching Degree.org has compiled a list of 100 Helpful Websites for New Teachers, and the list includes new teacher "survival" sites, teacher video sites, freebies, inspirational sites, and several other categories as well.

Some Math As Well

At Exploring Binary blog we find an EXCELLENT way to explain binary numbers that he found when he taught his mother binary numbers . It's true, once you struggle at explaining a concept to someone who's not a math whiz, you really get to the brass tacks of the matter.

Denise from Let's play math! has some geometry challenges for us, some of which are from ancient Egypt. I solved the puzzle under the title "Can You Explain This?", using fractions (and not avoiding them as it tells you to do). I also pondered number (2) but couldn't find an immediate (easy) way to find an answer. I don't think she ever posted answers to those.

John Cook from The Endeavour blog shows us an optical illusion that is so fooling that I couldn't believe my eyes at first. Quite amazing. He also makes an analogy from the optical illusion to a "mathematical" illusion within his own work.

Dr. Jeff is asking us to fold a humongous piece of xerox paper and see how many folds reach to the top of mount Everest or the sun, among other things. You probably won't believe the answers!

If you're into 5th-7th grade math, please check also my videos about integer subtraction.

---

This is where to send your submission for the next carnival on July 10th at Math Mama Writes. Happy summer, everyone!

Wolfram|Alpha is here

Wolfram|Alpha is a new, computational search engine. If you do a query where the answer has quantitative data, Wolfram|Alpha probably gives it.

For example, try enter your last name. Wolfram|Alpha gives you information about the popularity of that name. For example, "miller" ranks 6th within the US and there are about 1,128,000 people with that surname.

Enter a first name, and it will even give you a graph showing the name's popularity over the years. I just found out that Cindy's popularity peaked in about 1960. No wonder it is a common first name among the mothers who ask me questions about their children's math education.

Enter a town - for example Houston, and see what information comes up.

But, the reason I'm writing this post is because Wolfram|Alpha (or Walpha as some called it) especially excels in mathematical queries.

This has implications to mathematics education, especially in high school and college. This tool is completely free, easy to use, and accessible for everyone with an Internet connection (even with a smart phone). Just imagine if you are a math teacher and you assign homework for your 9th graders, "Find the equation of the line that goes through points (2,5) and (-8, -9)." W|A does it in a split second.

It also solves equations. For example, Solve ln(x)+ ln(x-2)+ ln(15). It even gives you the solution steps - just click on "show steps".

Please see a full list of examples of what Wolfram|Alpha can do in mathematics.

While W|A easily computes a lot for the elementary algebra or calculus student, it doesn't stop there. Look at the examples for elementary mathematics given on the site. It also acts as a fraction and percent calculator. Granted, normal calculators do those as well.

So, what is a teacher to do? Will a lot of the content in high school and college math courses suddenly become obsolete?

Now, Wolfram|Alpha is not what started bringing technological tools into classroom. That trend has been here for a while (think graphic calculators, computers). However, it is an extension of the trend that makes the computational tools more easily accessible and available to nearly all students.

I see Walpha both as a benefit and as a drawback.

- BENEFIT: Teaching can focus more on the concepts and less on tedious calculations, since practically all students will have a tool they can use for computations and graphing (all you need is a computer with an Internet connection or a smart phone). Students could be given more problems of the type that require thinking and problem solving, and less of the mechanical calculation problems.

- DRAWBACK: It is a fact that one cannot adequately learn mathematics without also learning many of these "mechanical tasks", such as how to find the slope of a line when given two points, or how to graph a parabola from its equation. If these skills were totally skipped, students wouldn't be prepared for the next mathematics course where such knowledge is needed. So, teachers cannot skip those topics and need to enforce student learning in such a way that cheating with Wolfram|Alpha cannot happen (putting such problems in the test).

Actually, I'm not so sure if W|A will change the actual teaching so much, because many students have already been having graphic calculators, which do similar things as W|A. I wonder if the biggest change it brings along is a decline in the sales of graphic calculators...

Read also what others have blogged about W|A :

Impact of Wolfram Alpha on Math Ed

Wolfram Alpha is up and running

Wolfram|Alpha and the shrinking future of the graphing calculator

Walpha Wiki discussion

Measuring worksheets

I finally got around to creating another worksheet generator that had been "lacking" for a while from my "collection" - measuring units worksheets. Those where you ask kids to convert 6 cups to ounces, or 5 kg 40 g to grams.

Measuring units worksheets

These are free and printable. I've made lots of ready-made links for some common type worksheets, but of course with the generator you can customize it how you want. Both metric and customary units are included; however I did not include all possible metric units, because the focus here is for grades 3, 4, and 5.

Integer subtraction

I recently finished another video of mine, this time on the topic of subtraction of integers. You can watch it here:

Subtracting Integers

In it, I explain three different models that we can use to justify the rules for subtracting integers to the students. The three models are:

1) number line jumps;
2) concept of difference;
3) counters.

Please read about these models in more detail in my updated article How to teach operations with integers.

Fraction games online

I updated my list of online fraction games and tutorials, and sorted the games according to major topics such as equivalent fractions, fraction addition & subtraction, multiplication & division, or fractions & decimals. Take a look!

Massachusetts teachers' math exam

Photo by Made In China

I was almost ready to comment on this exam, where only about 27% of aspiring aspiring elementary school teachers passed the new math section of the state's licensing exam this year...

Boston.com says about this test, "Education leaders said the high failure rate reflects what they feared, that too many elementary classroom and special education teachers do not have a strong background in math and are in many ways responsible for poor student achievement in the subject, even in middle and high schools."

...and then I noticed I had been looking at the wrong link for the practice test. The real link is this:
Massachusetts Tests for Educator Licensure, Mathematics Subtest, from MTEL Practice Tests website.

I feel that test is pretty good! In fact, I'd recommend that you do some problems from it, if you're teaching any grade from 1-12. If you're teaching middle or high school, you could use some of those problems with your students, and if you're teaching elementary, it's just to check if YOU have the adequate math skills.

The open-response item is particularly interesting, and good, I think. It shows a student response on a particular geometry problem, and asks:

"Use your knowledge of mathematics to create a response in which you analyze the elementary school student's work and provide an alternative solution to the problem. In your response, you should:

• correct any errors or misconceptions evident in the elementary school student's work and explain why the response is not mathematically sound (be sure to provide a correct solution, show your work, and explain your reasoning); and
  
  
• solve the problem using an alternative method that could enhance the elementary school student's conceptual understanding of ratios and decimal multiplication in the context of the problem."

Take a look at the math test.

---

What kind of math would I test elementary teachers on

If I made a test for future elementary school teachers, I'd ask lots of questions about elementary and middle school math INCLUDING "WHY" questions. If they know that, then they can explain the math to their students as well.

I might ask questions related to common errors and misconceptions kids have. "Sally calculated that 0.5 + 0.12 = 0.17. What concept is Sally not understanding (and it's not decimal addition)? What kind of intervention do you think would help?"

I would ask questions that test their understanding of why long division or long multiplication works.

I'd test their knowledge of middle school level math and some high school level math. I'd test for problem solving abilities.

Elementary teachers should know middle school math well (percents, proportions, equations, geometric constructions, statistical graphs, etc.) so that they know what the elementary math they teach is leading to. For example, they should know square roots and Pythagorean theorem. That way, when they teach multiplication, they can throw in a "teaser" for the best kids in the class, asking, "What number multiplied by itself is 64?" Or, "I say a number, you say what number multiplied by itself gives that number."

So, I might actually ask, "What further mathematical concepts after 3rd grade depend on a good knowledge of multiplication tables?" Or, "Students study prime factorization in 6th grade. Give two examples where the understanding of this concept is needful in further mathematics studies within grades 6-12."

See also what MathMama writes about Tests .... (TIMSS & the MA teacher licensing test)

Word problem Singapore way

Laura had 24 clips more than Holly. After she gave 5 clips to Holly, Laura had twice as many clips as Holly. How many clips did Laura have left?

This is from Singapore Challenging Word Problems book 3 (which they are now discontinuing, I heard).

I found two ways to solve this using the bar diagrams.

Solution 1. Notice that this is showing how I solved it initially, and at one point I had to adjust the length of the bar.

---

Solution 2.



In either case, once you get that x = 9 (the amount of clips Holly had in the beginning), it's easy to solve that Laura had 28 clips left after giving 5 to Holly.

---

Algebraically:

Initially Laura has L, Holly has L - 24. (Obviously you could also choose to use H and let Laura have H + 24.)

Then Laura gives 5 to Holly, so now Laura has L - 5 and Holly has L - 24 + 5 which is L - 19.

At this point Laura has double as many clips as Holly: L - 5 = 2(L - 19) and we can solve.

L - 5 = 2L - 38
L = 33.

Laura had 33 - 5 = 28 after she gave 5 to Holly.

There are probably other ways to solve this as well.

Math Teachers at Play #8

Go visit Math Teachers at Play carnival at Let's Play Math. Lots of neat stuff this time!

I definitely want to try this with my daughter for multiplying by 7s. I also really liked
Four things I used to think about calculus, and what I’ve replaced them with
- a teacher whose teaching has evolved towards conceptual understanding.

Research on conceptual understanding

Just an interesting piece... a recent study has found that teaching conceptual understanding in math makes children learn better, as opposed to teaching procedures.

You Do The Math: Explaining Basic Concepts Behind Math Problems Improves Children's Learning

The children were taught about solving equations such as

4 + 5 + 3 = ___ + 3

either procedurally, or conceptually. Procedural instruction went kind of like this:

"Add the three numbers on this side, 4 + 5 + 3. That's 12. Then subtract the number here from that, 12 - 3 = 9. That's the number that goes to the blank."

In the conceptual group they were taught about equivalence, the equation having two sides that have to be equal.

Of course... the conceptual teaching is the way to go!

Here's a link to the research paper (in press):
Matthews, P. & Rittle-Johnson, B. (in press). In pursuit of knowledge: Comparing self-explanation, concepts, and procedures as pedagogical tools. Journal of Experimental Child Psychology.

Recommendations for pre-algebra

Folks often ask me how far I'm planning to write my Math Mammoth complete curriculum (the Light Blue books). Right now I'm finishing up 5th grade. After that, I will be writing 6th grade. But, I'm not really planning to go on after that.

I feel a child who will have finished Math Mammoth 6th grade (once it is available) is probably ready for pre-algebra for 7th grade. Perhaps not all children won't be, but a good portion of those who use Math Mammoth should be.

Pre-algebra is sort of "in-between" course. It is bridging the world of numerical computations of elementary math, and the world of algebra where we manipulate variables. Pre-algebra courses typically cover these topics:

Photo by ecumaniac

* integers
* fractions, decimals
* factors, exponents
* solving linear equations
* solving linear inequalities
* ratio and proportion
* percent
* graphing linear functions
* Pythagorean theorem and other geometry topics
* some statistics and probability.

Most students take prealgebra in 8th grade, just prior to taking algebra 1. However, it is possible to study these topics in 7th grade as well, if the student has a good foundation from grades 1-6. And that is what I'm aiming for in writing Math Mammoth - a good enough foundation of concepts so the student can go on to pre-algebra after 6th grade.

I have prepared an article that talks more about the various pre-algebra book choices.

Newest Blog Carnival is online

Check out the latest edition of Math Teachers at Play blog carnival. It's up at Homeschool Bytes. As a carnival pick, I enjoyed Plat Diviseur (Fractions on a plate) post. It's about a dining plate... Go see!

Long division and dyslexia

Photo by laura.bell

I have a dyslexic 9 year old and I wish to find out if your program can benefit him. I have tried Math-U-See and it became too boring or frustrating for him. I have tried various other programs in hope of not overwhelming him. He has completed the delta Math U See level but I feel needs more work on long division. He simply gets very frustrated with it, as the length of time and knowing where to place number due to dyslexia. Is your curriculum a spiral curriculum? Any suggestions would help.

One of my books from the Blue Series goes through long division in several small steps: Math Mammoth Division 2

I would suggest that for a dyslexic child, have him do ALL the problems on a squared paper (grid paper). That will help him place the numbers right. Not all the problems in my division book are done with the grid... but for your son, it may be necessary to always use such paper.



Secondly, when you teach on board or on paper, at each step COLOR the whole column of ones, or tens, or hundreds (whichever you are working at). I have not done such exactly like what I explain now in my book, I'm just telling you to try that: color the column you're looking at, at each step.

This will help him focus on the specific place value, such as hundreds, and help him place the digit in the quotient in the hundreds column, write the product, and calculate the difference in that column.

Apart from those tips, it might also help if you check whether he understands the REMAINDER concept outside long division. For example,

16 / 5 = ??

42 / 10 = ??

He must be able to do those well in order to some day understand why long division works. HOWEVER, it's possible for children to learn the motions of long division without understanding why it works. So, definitely do not discontinue long division just because he doesn't grasp why it works. That understanding may come later on.

My curriculum does not use a "short" spiral like Saxon/Abeka/Horizons. It is more mastery based. However, different concepts are reviewed and studied usually on 2 or 3 neighboring grade levels, and I also use problems about new concepts that also use previous concepts so they cannot be forgotten. For example, once they learn about writing addition and subtraction from the same picture, then that is used to learn fact families, which is also used later on with x.

Hope this helps.

Chocolate and mental performance

Photo by Cacaobug

You might have seen this piece of news about a new study:

Scientists reveal how eating chocolate can help improve your maths
Eating chocolate could improve the brain's ability to do maths, a new study suggests.

I think I have experienced this effect. Whenever I eat several pieces of dark chocolate, it definitely seems (while I do various computer work) that my brain works faster and I'm more alert. I also think that this effect is not because of sugar, because the dark chocolate does not have high amounts and because I definitely don't feel the same if I just ate, say, a sweet apple or a cookie.

I like both chocolate and the "alert" feeling after eating it. I don't like it when I start feeling the "brain drain" (your brain can definitely get tired after several hours of researching/writing emails & blogposts/devising math problems etc.). I don't know if this effect is from flavanols in chocolate like it mentions in the study, or theobromine, or both, or also something else.

The brand I often get here is El Rey. I don't have a clue about its flavanol content; it says it's made with "rare, flavorful, coveted" Venezuelan cacao beans only. But be that as it may, the taste sure IS good!

Math Teacher's Carnival

This week's edition is posted at I Want to Teach Forever blog.

CurrClick has a Spring sale

Discounts vary up to 75% off in CurrClick's Spring sale for all kinds of downloadable products (this includes all school subjects).

Math Mammoth titles at Currclick are discounted too.

Paper models for polyhedra

Have you ever wanted to fold a pyramid or some of the other interesting polyhedron out of paper?

At least, take a look at the pictures! www.korthalsaltes.com provides hundreds of free printable paper models for various solids, including platonic solids, pyramids, and lots of polyhedra. Some are available as colored versions also.

Folding and gluing one or two is a great summer math project also, and will keep kids busy for a while.

Singapore math approach marches on.. to Houghton Mifflin Harcourt

Houghton Mifflin Harcourt has just announced the launch of a new math curriculum, titled "Math in Focus", which is adapted from Singapore's most widely used program, My Pals Are Here! Maths.

Who would have thought! Not I. This series contains "carefully... paced instruction of fewer topics at each grade in greater depth to ensure mastery". That's exactly what I've wanted to see in school books and what I have tried to create with Math Mammoth.

It also has a close alignment to the NCTM Focal Points, a "concrete to pictorial to abstract" approach, and instruction centered around problem solving using multiple models.

I feel quite surprised. See Math In Focus website for more.

It appears NCTM's Focal Points are having their effect.

A retired school teacher's comments

I received this comment a few days ago concerning my Coherent Curriculum article, and I thought it's enlightening enough to post here as well (emphases mine):

I am a retired public school teacher. I taught first, third, and 1-5 self contained LD and resource classes 1-5. As I walked the halls of my school and heard the lessons being taught, I was so enraged! I knew students sitting in those classes that had learned those skills I still heard being taught and retaught. The work on display outside the classrooms showed very little improvement beyond what the same children had done in the earlier years.

We spin our wheels over and over teaching the same thing year after year and wonder why they can't do better on testing. They are bored to death. This is not just in math. It is in phonics, language skills, creative writing, basic logic skills. As a nation we teach a "swallow and spit back out" curriculum that is mandated by the state standards.

If a child is capable of going beyond, teachers are not allowed to take them higher. "It's not on the grade level standards". Likewise, if a student needs to review last year's standards, it is not allowed. We must stay on grade level standards.

Now do you see why Johnny can't read, spell, write, solve problems, of think. He never had to. He just memorized information for a test and never saw it again until the next year when he rememorized the same facts and passed with flying colors. Homeschool teachers, please don't teach the "memorize and spit back out" method. Make you children reason, explain why or why not, defend a position with research data. Don't put out robots. Public school has already handled that job very well.

Jeanette

Teaching negative exponents

Continuing on with the exponents, I have another video about negative exponents. Check also the "prelude" to this, my video on zero exponent.

Negative Exponents: Learn Them with a Pattern!

In books we usually find the definition that x⁻ⁿ = 1/xⁿ or perhaps x⁻ⁿ = (1/x)ⁿ (you take the reciprocal of x and use the opposite of the exponent).

But again, I want to highlight that we DON'T have to just "announce" to students this definition or how negative exponents are done. We can justify it. In the video, you can see this done first with the help of a pattern, where we go from the positive exponents to zero exponent to negative ones.

I also show another way of showing your students why negative exponents have been defined the way they are. If you look at the shortcut for dividing powers with the same base (I used 6³/6⁵ in the video), we can simplify as usual and arrive at 1/6². However, the shortcut says we can just subtract the exponents, so 6³/6⁵ also equals 6⁻². The same reasoning applies for any non-zero x and any positive whole number exponent.

This way, we teach our students not only the "how" but also the "why" of mathematics, and help them become better thinkers. Mathematics is not just about memorizing things and then learning to "spit" them out at tests - it also has a LOT to do with proof and logical thinking. And surely our kids need that today, more than ever.

Zero Exponent - with a Pattern!

Many school books just "announce" the rule or the fact that any number to zero exponent is one (excluding zero to zeroth power, to be exact). I like to call this "announced" mathematics - it's math without justifications, without explaining the "whys".

In the video below, I show you a better way, where we approach it through a simple pattern. This justifies the rule and makes it logical, instead of just a piece of "announced" mathematics without proof.

Zero Exponent video

I also show another way of justifying it, which has to do with the shortcut for multiplication of powers with the same base. For example:

Following the shortcut for multiplying powers with the same base,

x⁷ x⁰ = x⁷, because you can add the exponents 0 + 7 = 7.

This is an equation where x⁷ is multiplied by x⁰, and the result is x⁷. What must x⁰ therefore be? It must equal one.

See also my earlier article on this topic: Negative or zero exponent.

Blog Carnival

As I seem to be lacking creative ideas for blogposts on my own blog (though I'm working on two new videos), I suggest you head to Math Teachers at Play blog carnival.

Lots of good stuff there! For example, MathNotations is holding a second (free) math contest, and I found some neat Russian-style "subtraction" problems.

As always, Denise has illustrated the carnival beautifully.