We are all “Math People”!?

Consider, for a minute or two, that the beliefs that we are each either “a math person” or “not a math person” are holding back our potential achievements. Are there things you would do differently if you were a “math person”? Could you discover something new about mathematical relationships if you weren’t taking it for granted that you are a “math person”?

It seems that on either side of the math person belief system there are people who limit their understanding by holding fast to these ideas. The students who are “math people” tend to stop trying, they believe they already have it. When a new problem becomes challenging they tend to give up, and reconsider being “math people”? The students who are “not math people” believe they will never get it, so they do not even try. These beliefs are the accepted standard in our country. When I mention taking a math class to an acquaintance I undoubtedly receive an “I’m sorry” or “that stinks” response.

 

Now consider that these “math person” beliefs are not just holding back our academic potential, they are limiting our brain’s growth. Practice and hard work can actually  change our IQ scores, for a time. Research shows that problem solving increases connections in our brain, which improves brain function, ultimately increasing IQ scores. When we stop practicing (or using our mind power) these connections slowly weaken and IQ scores can go back down over time. So, keep your mind active! Just like other muscles in our bodies, our brain needs exercise. One way to exercise our grey matter is critical thinking about mathematical word problems. It is crucial that students are given complex problems, and time to develop solutions on their own.

This does not mean that teachers can hand out word problems and sit back while students “discover” the answer on their own. Before students can explore math problems through critical thinking, they must be taught how to do so. Fortunately George Polya developed a process with four easy to follow steps for this purpose.

 

Polya’s 4 Principles for Solving Problems:

  1. Understand the problem, consider the wording and information
  2. Devise a plan, choose an appropriate strategy
  3. Carryout the plan, be persistent and willing to redo work
  4. Look back, check your result, do other strategies work? does this apply to other problems?

 

A variation of Polya’s process named SQRQCQ was developed by Leslie Fay in 1965:

  • Survey. Read the problem quickly to get a general understanding of it.
  • Question. Ask what information the problem requires.
  • Read. Reread the problem to identify relevant information, facts, and details needed to solve it.
  • Question. Ask what operations must be performed, and in what order, to solve the problem.
  • Compute/Construct. Do the computations, or construct the solution.
  • Question. Ask whether the solution process seems correct and the answer reasonable.

Fay’s process, SQRQCQ, may be even easier for students to work with. I like how it clearly emphasizes questions in each step of the process. Reflecting on our progress by asking questions models the way we want students to be thinking. These processes are useful to all students, and are important in helping special ed students succeed in Math.

Finally, consider the empowerment that comes from an “I can do it” attitude. Show students that they can improve their math skills with practice. Talk to them about how our brains work, help them understand that intelligence is not something we are born with or without. Mathematical wisdom comes from experience, and it is worth working towards. Help to put an end to the “math people” myth.

Understanding Rational Numbers, Fraction Fun

The set of rational numbers includes all natural numbers, whole numbers, integers, and fractions. A rational number can be represented by any one of its equivalent fractions. This can be an extremely overwhelming concept for students (and for teachers too). As I have said in previous blogs, the most meaningful path to understanding (or math literacy) is through real world situations and making the computations visible. Fortunately for teachers, there are many wonderful resources available for making fractions fun!

There are infinite food objects that can be divided and shared to illustrate fractions, pizza being a popular choice. There are paper cutouts and color a section of this shape worksheets. These are all great tools to help students visualize and understand equivalent fractions. As a teacher, I will use all of these strategies, because fraction practice in a variety of ways will increase deeper understanding by my students.

I am a believer in the power of physical activity to increase brain activity and comprehension. That is why I look for ways to be active in the classroom. If a concept can be explored through movement, I will get my students moving! Right now, my favorite form of fraction discovery is bowling. It is easy to implement, a great way to see those fractions, and fun for students. Many cool worksheets for this activity can be found online, or you could make your own.

To play fraction bowling, you need: a number of pins (5-10) for each group (I recommend 2-3 students per group), and a scorecard for each student. Have the students make fractions to represent how many pins they knocked down, and how many pins they missed. Ask questions about equivalency like: which is more 5/10 or 1/2?

For extended learning, change the number of pins that students start with. Compare the new fractions to the original fractions. Ask questions about the new fraction equivalencies like: what is a better score 5/9 or 5/10? How can we compare 5/9 to 5/10 to see which is more? Hint: find the common denominator.

Ok, bowling is fun and requires movement, and that is great for engaging students. However, in this digital era in which we live, teachers have to embrace technology. I realize that technological devices are an important part of my student’s (and my own) lives. Luckily, there are really great online tools for working with fractions! Kathy has created a page with links to interactive fraction websites. I recommend Fraction Bowling as a follow up to the classroom bowling activity. There are a lot more ways to make fractions fun, so why not engage, inspire, enjoy!

Decomposing Numbers, I didn’t learn this in school!

 

When I was an elementary student, we were taught the traditional algorithms for mathematical computations. Which, for addition involved “carrying the one”. To solve 127+89=___ we would write:

 

127

+89

 

and then compute the sum using columns and “carried ones”. Like this:

    11      

127

+89

216

This method worked well for me. I understood that 7+9=16 ones, and 6 would be written into the ones place (now called units) for my total sum. I understood that the 1 from the 16 needed to go with the other tens, and thus 1+2+8=11 tens. I put the 0 into the tens place of my sum, and  “carried” the 1 into the hundreds column. To finish the computation I had 1+1=2 hundreds and my total sum of 216. Yay for me, I found traditional addition methods easy to work with. I was using number decomposition in my method, without knowing it.  So, how will I help my students who find the standard algorithm a little difficult? The ones who believe they are “adding one” to the tens column rather than adding ten. The answer is, make the sum visual.

 

There are many ways to make what is happening during the addition process visual! What they didn’t teach me in school is that my thought process can be written out using the partial sum algorithm. Actually writing out the decomposed numbers creates more understanding of number and place value relationships. I was, at first, skeptical of the value of new math teaching strategies, but now that I have used them I can’t get enough.

 

0505161933

An example of sorting (decomposition). My son composed this arrangement of flowers, all his idea, on his own.

Solving my original problem with the partial sum (or instructional) algorithm would look like:

127                                                                 

+89                                                                 

(7+9=) 16                                                                 

(10+20+80=) 110                                                                

(100+100=) 200                                                               

216                                                               

 

Each number can be decomposed so that the sum reads:

 

127+89=(100+20+7)+(80+9)

 

to make the addition easier, we add the similar numbers first, and then find the total sum:

 

(100)+(20+80)+(7+9)=(100)+(100)+(16)=216

 

I love the simplicity of decomposing numbers for problem solving. It gives students a clear, and easy to understand, way of thinking about large numbers. It seems to come naturally to kids, as with my son and his flowers. He picked them and sorted them by size all of his own design. He spent time looking for the ones that fit between. He just sensed a logical progression. To take this thinking further pictures and manipulates can be used. There are many resources available to teachers for helping students visualize unit measurement. One source that I like is: Ed Helper. They offer printables like the one below. You need a membership to use their pages, and there is a small fee.  

example

 

Math Literacy – beyond terminology

An important element of critical thinking on any topic is literacy, or competence. Math is no exception. In order for students to gain a deeper understanding of mathematical concepts, they must first become math literate. This means that students can apply mathematical reasoning skills to help them solve real world problems. It sounds like a daunting task, but it doesn’t have to be so grand. For young students math literacy involves knowing if they should use addition or subtraction to solve a word problem. This can transfer to the real world experience of biking to the corner store to purchase a treat, and making sure that the correct change was received. Older students will have more complex real world experiences, such as, earning enough money to purchase tickets for a concert. They can be encouraged to set up equations while problem solving.

Our job as teachers is to ensure that each student becomes math literate. The first step is to promote an expectation of success in math for all of our students. We have to believe that our students can do well, and then they will believe it too. Set up a culture of critical thinking on a variety of numerical topics. Invite students to create their own math problems, and to solve them. Set aside some class time for a discussion of thoughts about math. Tell students that there are no bad ideas, and watch the discussion grow. Even if no actual math computations are made, the discussion will reinforce the idea that math is important and that everyone can be good at it with practice.

Students need to understand that the mathematical language is important in conveying ideas, but math literacy goes beyond knowing terms. It involves a deeper thinking, the ability to view problems from multiple perspectives, and to apply previous knowledge to new situations. Giving our students the time and the skills to develop competence in math gives them the confidence for continued academic success.