What's Math Got to Do With It - Intro and Chapter 1

In the introduction to What's Math Got to Do With It, Boaler notes interesting trends - many adults say they hated math in school, but like math in their work. Many adults enjoy activities like Sudoku, that require logical thinking, but they did not enjoy their math experiences in school.

The math that is needed for working situations is logical thinking, comparing numbers, analyzing and reasoning. People need to be able to reason, problem solve and apply methods to new situations. An official report examining math needed in the workplace revealed that estimation is the most useful math activity.

Boaler references Conrad Wolfram's TED talk in which he talks about math as a four-step process: posing a question, constructing a model to help answer the question, performing a calculation, and converting the model back to the real-world solution by seeing if it answers the question.

Boaler shares that her book will identify the problems that American students encounter and will share some solutions.

In Chapter 1, Boaler talks about the difference between how students view math and how mathematicians view math: students see math as "numbers" and "rules," but mathematicians see math as "the study of patterns" or a "set of connected ideas."

According to Boaler, math is a "set of methods used to help illuminate the world." She discusses the Fibonacci sequence and the golden ratio, which many middle and high school students have never heard of.

Boaler discusses quite a few differences between "school math" and "mathematician math." Mathematicians work on long and complicated problems that involve combining many areas of math, while school children spend hours answering short questions that address the repetition of isolated procedures. Long, complicated problems encourage persistence, and an important part of "real math" is the posing of problems. According to Boaler, mathematics involves going from answer to question, while computation goes from question to answer.

The work of mathematicians is collaboratory. Mathematicians do not work in isolation - when interviewed, mathematicians have stated that they collaborate to learn from one another, increase the quality of ideas, and share the "euphoria" of problem solving. However, Boaler states, there are still many silent math classrooms where students work in isolation.

I enjoyed reading about the "meaning" of math, and I will pose the question to my students this week - "What is math?"

Reading this chapter has inspired me to create a couple posters that reflect what math really is.


Why I'm Not Teaching Decimal Operations "Rules"

Math rules. How often have you found that students are taught "tricks" to remember math rules? How often do they make procedural mistakes even though they've "learned" the rules?

I have taught decimal operations for more than 20 years, and I have seen, time and again, students who know how to add and multiply decimals but then follow the wrong "rule" for the operation they are completing. Line up decimal points when adding or when multiplying? "Jump" the decimal point over when adding and subtracting? Or is that multiplying? They don't remember when to use which method to place the decimal point.

So, this year, as we approach the decimal unit, I've been feeling like I don't want to talk about the rules for where/how to put the decimal point. I want to focus on logic. Today that feeling was reinforced when I asked my students to solve 35.2 + 7.489 and then explain why their answer made sense. Here are some of the answers and reasons (I didn't teach this yet, but they learned it last year):

"0.11009 makes sense because I tried my best and if I remember correctly, addition problems you don't need to line the decimals together"  

"0.7838 makes sense because when I added I knew that it doesn't matter how it's lined up"

"78.42 - I added 9 and 2, then 1, 8 and 5. Next I added 1, 3, and 7. Finally I added 7 and 0 and I put the decimal in the middle."

"7.841 makes sense because with adding you only have to add the decimals on the top. Then you add and finally add the decimal back in."

"426.89 because I put the decimal point four spaces back because there are four numbers behind it"

 "79.41 makes sense because you do it just like an addition problem (that's how I remember it anyway)"

"7.841 makes sense because you add like normal and take the decimal from the farthest out and put it with the answer" 

A few correct answers, with reasons:
"42.689 - this makes sense to me because this is how I learned it. You do simple addition, but line up the decimal points"

"42.689 makes sense because I used what my fifth grade teacher taught me, line up decimals, add zeros so everything is lined up and then solve."

"42.689 - I don't know how it makes sense, but it's how I learned to do it." 
Of the 120 students in my classes, only 8 said the answer made sense because "35 + 7 is 42" or because "I estimated" or "when we're doing addition, we know we end up with a bigger number."
Now, that doesn't mean that they didn't think about those things, but to them answers seemed to "make sense" when they followed the rules - even if the rules are remembered incorrectly; students got right and wrong answers and they all made sense because that's "how they learned it."

So, what is the point of teaching rules? Especially to those students who are a little weaker in math - if they can't remember the right rule, they can't tell if their answer is reasonable. They need to develop their number sense.
In the past, I have asked students to estimate the answer first, so they know if their answer is reasonable, and I have required them write these estimates on their tests. But we've also talked about the rules. I'm thinking that if I take the focus off the rules and put extra focus on the estimating/reasonable answer idea, students will be better able to identify reasonable answers and will feel less dependent on the rules. 

I know that multiplication and division logic will be more difficult. Problems like
 23.5 times 4.428, won't be as bad because there are whole numbers involved. This could be estimated as 25 times 4 = 100. So when placing the decimal point in 104058, it should be placed so that the answer is about 100 - not 10, or 1, or 1000.
Now multiplying 23.5 and 0.7 may be more confusing, but this will be the time to help them understand why the answer should be smaller than 23.5....but more than half of 23, since 0.7 is more than 0.5.

Go to YouCubed to download this awesome activity!
I think division will be the most challenging, as far as determining reasonable answers, and I need to think about this one a bit more. However, we have already done this activity I found on YouCubed -"Too Big or too Small Maze Board." In attempting to create the largest number possible (using a calculator to compute), many students have already made the discovery that dividing by a number less than one gave them a larger number, while multiplying by a number less than one gave them a smaller number. No rules were taught - they found this "secret" on their own. This will be great to reference and discuss when we begin working on the multiplication and division of decimals.

We'll see how it goes!


Math Mistakes Lead to Brain Growth

I've been reading the book What's Math Got to Do With It, by Jo Boaler, published in 2008 and republished in 2015. When I started reading, I was a little disappointed that I hadn't found the book sooner - there is so much interesting information!

The preface alone captured me, because Ms. Boaler explains that, based on recent brain research, anyone can learn math to high levels. She has created an online course, How to Learn Math, for students to learn about how brain research supports the idea that anyone can learn math - being good at math is not a genetic predisposition.

She also discusses a site she created called YouCubed, which I've been using to find tasks for my students to work on. (I just visited the site and saw that she has a new book out  - Mathematical Mindsets! I must get it!)

Boaler references Carol Dweck and her book, Mindset: The New Psychology of Success (I'm reading this one too), which is based on the idea that those with a growth mindset, who believe intelligence can be learned, reach higher levels of achievement, engagement and persistence. This concept, to Boaler, implies great possibilities in the area of math and math teaching.

An interesting idea stated in the preface is that people being "naturally good at math" (or not) is a myth that is "strangely cherished in the Western world but virtually absent in Eastern countries" that top the world in math achievement. In classrooms that Boaler observed in China, numerous high school lessons focused on no more than three questions in an hour - students did most of the talking and discussing and they worked on these questions in great depth....quite different from many classrooms in the US.

Boaler references research that reveals that the brain can rewire and change and grow in a short period of time. She states that while traditional math teachers believe that some students aren't able to work on complex math, it is that very working on complex math that actually enables brain connections to develop. If students who are considered "low" are given low-level tasks, they won't develop those brain connections.

Boaler points out an idea that I hadn't thought about - math is often viewed as a "performance" subject, rather than a "learning" subject.

Other interesting research Boaler shares - when a student makes a mistake in math, the brain grows. The research shows that making mistakes causes synapses to fire and connections to form - even if the student isn't aware of the mistake! This is because making a mistake is often a time of "struggle," and brains grow the most when they are challenged and engaged with difficult work. When one becomes aware of making a mistake, the brain "sparks" again. Based on this idea, Boaler recommends that students be given math work that does create the struggle, rather than structuring lessons to allow students to arrive at correct answers most of the time.

I've read the first few chapters, but I'll reread to decide what to share here - check back if you're interested!

Has anyone else read this book?


Problem of the Week #7

Happy Tuesday!

I got up at 4:30 this morning (again) - because my dog decided to start barking (again). He's getting older and has started waking up pretty early most days!  So, instead of trying to get a little more sleep on the couch after I took him out, I figured I'd work on my problem of the week!

This one is an area/perimeter-related problem, with answer key included.

Click to download!


Problem of the Week, # 6

I hope everyone has had a relaxing weekend and is feeling ready for the coming week. I'm feeling a little more ready than usual:-) This weekend, I enjoyed watching my daughter's band competition, spent a little downtime reading Mindset, by Carol Dweck (and ate a bit too much chocolate!). I've heard much about the book, and am finally reading and enjoying it.

The Problem of the Week this week is another logic puzzle - hope  you like it!

Click to download


The Benefits of Self-Correcting Math Work

I have been teaching for more than 20 years. If you have been teaching for a long time, then like me,  you may have used a certain strategy/instructional tool for a period of time, and then for some reason, stopped using it....and then after another period of time you came back to it, and wondered WHY (or when!) you stopped in the first place!

That was me today. I had made 20 copies of my Footloose answer key and had the students correct their own papers (they had worked on the Footloose activity for part of yesterday's class and then finished during today's). I was surprised by the thoughts that went through my brain as they were correcting - the main one being - "When did I stop doing this?!"
I do have students check their homework answers with the answers shown on the board (sometimes), but I don't give them each a detailed answer sheet to use, and I rarely have them grade their own classwork.

Here are my re-discoveries related to students correcting their own math work. Some of these may be particular to the topic we worked on (writing algebraic expressions from phrases, phrases from expressions, and evaluating expressions given a value for the variable), and the fact that the answer keys were detailed (not just the answer), but I'm sure I'd observe the similar things when studying different topics as well:
1) Students asked me more questions when checking their work with my key. Since they were working at their own pace and checking individually, they seemed to be more comfortable with verifying whether or not their phrases were ok (I didn't have every possible phrasing option on my key). Students who wouldn't normally raise their hands to ask in front of the class did ask me questions during this time.
2) Correct work is modeled on the answer key. Because I had several options for phrases on my answer sheet, they had to read each one to see if theirs was on the sheet, giving them a little more exposure to correct options. I also had the steps for evaluating each expression, so they could go line by line and have those steps reinforced, as they compared the work to their own.
3) Students were finding their OWN mistakes, rather than me finding them. I heard things like, "I copied the problem wrong," "I said 3 x 3 was 6!" "Oh, I put division for product." And I realized, as I did years ago - it makes so much more sense to them when THEY see the difference between the correct work and the mistake they made, rather than ME finding they really know why I circle a mistake that they made on their paper if they don't take the time to ask me? When they find the mistake, they know what happened. I don't need to make those types of connections and observations. They do.
4) Students are engaged - they enjoy having the key! It was fun to see them with their pens or colored pencils, pointing at their papers, question by question, making sure they were being accurate in their grading of themselves, and then being sure to write the correct answer accurately (I did make them write the correct answers, using pen or a colored pencil, so the change would stand out).

I don't know what prompted me to copy the keys to use today, but I'm so glad I did. It's wonderful to be reminded of forgotten/lost practices that help students to think just a bit more.

Have you re-discovered any strategies/practices recently?


Problem of the Week, # 5

Fall leaves are looking so beautiful here! I hope you're enjoying the fall season, wherever you are.

This week's problem is a logic puzzle. I showed it to my daughter today (she's in high school) and was surprised to find out that she had never done one of these in her elementary or middle school classes! I know that I don't use these as often as I used to, so I'll be making more to use with my students...logic and reasoning are so important!

Click to download.
Or click here:-)

Previous weeks can be found here:  Week 1
                                                        Week 2
                                                        Week 3
                                                        Week 4



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