Critical Thinking Skills

In our instructional team meeting, we've been looking at data. Which is quite depressing. But what we've noticed is that basically, our students are not retaining information. The more we talked about it, the more we decided that the real issue is critical thinking. Students can only do exact duplicates of problems we've done in class without being able to apply that concept in a different setting. From there, we are brainstorming what we can do. Here's what we came up with as a group:
  • Include an entire unit of story problems
  • Use brainteasers/puzzles/riddles to get students in the habit of thinking and working multi-step problems in a 'fun' setting with no pressure.
  • More practice reading math problems 
  • More practice working multi-step problems
  • Teaching students to persevere and try different methods rather than giving up
 On a personal note, here are some ideas I want to try next year:
  • Error Analysis- Having students analyze work to find and fix the problem.
  • Journaling- Getting students to think through mathematical processes.
  • Peer editing- Have students read/assess each others journal entries.
  • Multi-Step problems- Make every problem more than one step if at all possible, including on assessments
That's all I can remember at the moment.

What are your suggestions? How do you  kick things up a notch? How do you write assessments that aren't just asking them to remember or memorize something?


  1. I would also like to include journaling in my classroom. How have you incorporated journaling in a math class?

    I planning to try more 'thinking' problems rather than 'doing' problems. Not quite sure what that will look like yet either.

    Last semester I did an entire lesson of 'discovery' that worked really well (very little lecture). I heard grumbles from many students because they actually had to think for the entire class period. It was solving linear systems using graphing, substitution and elimination. There were a few 'aha' moments when the student saw that the point of intersection was the same as the calculated point.

  2. I’ve been impressed with how Mimi at I Hope This Old Train Breaks Down creates classrooms that are more akin to an afterschool math club so my first suggestion would be to re-read her blog entries on how she sets up and runs her classroom. There’s a reason why she’s in demand internationally :-)

    Paul Hawking
    The Challenge of Teaching Math
    Latest post:
    March 22nd: WEAR RED . . . and maybe do more than just that

  3. Barbie,
    I haven't done a whole lot of journaling yet. The thing I do most often is having them explain the steps of how they solve a problem or list the skills needed for a particular concept.

    I tried some more general things last year like asking them to describe their perfect school, listing 10 reasons to be a teacher, and other such things. But now I have access to two books,
    Write to Know Nonfiction Writing Prompts for Geometry
    and Algebra
    that asks questions I really like. I've been talking with the English teachers about how to use them and they suggest as a assessment of what they already know before teaching a concept. But I like the idea of using them as a summative assessment as well. I don't know yet how it would work but I do know I will spend a LOT of time modeling.

    I've been trying to do as many 'discovery/investigation' lessons as possible which is a little easier to do in geometry rather than algebra. I'm doing systems of equations right now, would you like to share your discovery lesson? :)

    I am a reader of Mimi's blog but for one thing she teaches at a private school, and for another thing she teachers advanced students. I am a big fan of her scaffolded investigations and I've been creating as many of those as I can think of. Are there any specific blog entries you had in mind about how she sets up and runs her classroom?

  4. Yeah, I forgot Mimi now teaches advanced students at a private school, but at one point before that she taught at a public school in NYC that worked with students who were illiterate, had never done math before and came from broken homes. Can’t point you to any specific post, I’m afraid, but she does use labels, which is something I need to start doing, and she has 9 posts for the label “NYC”.

    Recommend watching this to get inspired for your coming adventures in developing future problem-solvers extraordinaire:


    One of Richard Rusczyk’s slides says this

    Why Challenging Problem Solving is Important
    -Presents challenges like the ones students will face in college
    -Prepares students for the intellectual challenges of many different careers

    His focus is on those incredibly gifted in mathematics, but a lot of what he says about problem solving should be true for all students.

    To the extent that you can, I would highly recommend developing practices that you can consistently follow the entire semester, rather than ones that are likely to get dropped mid-semester because of other priorities. I think your intentions with ACT practice suffered such a fate.

    One way would be to make error analysis a daily requirement of in-class practice. For some of my classes, after the day’s lesson I will assign two even problems, from the same page as the homework, for partners to work on. Partner A solves problem 1. Partner B then checks the solution. If it doesn’t check, then the partners have to figure out the error, correct the solution and re-check it. Then partner B solves problem 2 and partner A does the check. Then they turn it in to me to show that they actually did the work instead of goofing off for five minutes. An extension of this might be to make error analysis a requirement on assessments: you don’t show that you checked your solution by plugging the answer back into the original equation then you lose half credit.

    “C” students often do poorly for the simple reason that to their mind, doing homework is all about doing the problems as quickly as possible and then waiting until tomorrow when the teacher goes over homework to find out if they got it right or wrong. So it’s important, to me at least, that my students are forced to practice checking their work: my hope is that that daily practice leads to their checking their work on the test. I think that’s also a good habit to develop that will serve them well in later life.

  5. Paul,
    You are exactly right- I have to work on consistency in more ways than one. For error analysis, my textbook usually includes 1 problem per lesson where something is worked out or labeled incorrectly and students have to identify and correct the error. I love those but I need more problems for students to correct. I was thinking that if I used journaling as a form of summative assessment, students could peer edit each others paper as a form of error analysis. The four suggestions I listed are things that I think will be simple to integrate into my daily routine so that they are consistent practices.

    I also like what Amy does over at square root of negative one teach math by writing answers around the border of the notes page so students can cross them off as they use them. What an easy way for students to self check!

  6. I like the idea of error analysis, but am less fond of journaling as a component of math class. It tends to be a way to make math class more painful for those with dysgraphia or who are not native speakers of English, and gives the good writers a way to disguise their lack of math skills, without really appreciably improving anyone's math.

  7. @gasstationwithoutpumps: I understand where you're coming from, however, if I'm not mistaken, Illinois's end of course exam for algebra 2 has a decent-sized writing component. So how do you help students through that barrier to success without having them practice writing in the classroom?

    @misscalcul8: Related to my comment above, does your state mandate that the end of course exam score count for X% of their algebra 2 final grade in your class? Five percent seems to be the norm in those states where it is required to count.

  8. Gasstation,
    It seems to me that if students have weak math skills that writing would highlight that, not disguise it. Fortunately, I have all native English speakers. I feel like writing would help students under the concept rather than must the arithmetic being done. And I definitely think it is a critical thinking skill to even begin writing an explanation. Critcial thinking is my goal here. What suggestions do you have to improve critical thinking? I've noticed you usually disagree with my thought on integrating different things into the math class- what does your idea math class look like for increased student achievement?

    We don't take any end-of-course exam other than our individual final exams.

  9. @misscalcul8: Maybe I'm thinking of Iowa, or Indiana, or another state that begins with a vowel as far as state end of course exams go.

    I do think that you're on to something good with your plan to incorporate error analysis into your regular classroom practice. And my apologies for not stating that with my prior reply.

    An idea that came to mind is that you could have students write down the incorrect solutions they find and turn them in to you. Then during your chapter review day, you could put up the most common of these errors and have the entire class try to find them. In this way, students who reviewed error-free journals would still get exposure to common errors for each type of problem. (I wish I could do something like this but timing-wise, I need my partner A-partner B work to take 5 minutes or less and this could add another three or four minutes to the activity.)

    Do you have a sense of the types of writing prompts you want to assign in the journals? I'm still on the fence with writing math in the classroom and I'd like to get a better picture of what it would look like in your classes.

  10. Paul,
    If students are turning in their Practice A/B things to you, couldn't you collect them throughout the chapter and then throw them up on review day for students to various mistakes?

    Also, I made spreadsheets of prompts from the Write to Know series of nonfiction prompts. I like what I see, still trying to figure out exactly how to implement. Here is the
    Algebra Writing Prompts
    and the
    Geometry Writing Prompts

  11. I'd like to look at your prompts but I think you need to change the access from private to public (Google said they were restricted to more important people than me).

    Re: partner A/B work. I always have to plan an activity based on how long it takes the slowest pair to complete. Typically, I try to budget my lesson to end with 10 minutes left in the period. Even if I go over a few minutes, 9 times out of 10 that's still enough time to complete the partner work activity on a daily basis. With my current approach, all I get from the students is the correct solution and check--they erase the mistakes rather than rewriting the whole thing. If I have the partners spend a few minutes more to write the incorrect solution down as well, I increase the frequency of days where the students don't complete the error analysis. And I always feel it's better to have a consistent practice, even if you give up some benefit as a result.

    "A foolish consistency is the hobgoblin of little minds."

    Hopefully my hobgoblins are in check.

  12. Let's try again. Here's the

    What I'm envisioning is students work their problems, switch, the partner uses a different color to correct mistakes, and then they present it to you with the correct final answer. Maybe I misunderstood your explanation?

  13. Thanks for the working links. Need sometime to ponder over them, but it definitely gives me a better sense of what you're aiming for.

    For partner work practice, I've trained them for Partner A to solve problem 1 while Partner B watches. Partner B then performs the check and if it doesn't check, they both search for the error.

    Once they find the error, they erase it and correct the affected steps of the solution and re-do the check so that what's left on their paper is a correct solution and a check that worked. Otherwise, I could end up with a bunch of scratch out marks that make the solution unreadable and/or they'll write in the correct answer for the step they got wrong, solve it in their heads but not make the corrections to the succeeding steps on paper. Which means I will end up spending more time reviewing each of these papers to make sure they solved it correctly. (I'm basically wanting them to practice how they should check and correct their work on the test.)

    Instead, when they turn it in, I expect pristine easy to read solutions and checks for the two problems. It takes me a two second glance to see if they got it. If they didn't get it, or they come to me for help claiming that they don't know how to do it (which can be an excuse to get out of doing the work), I explain how to solve the problem and then send them back to their seats with another even problem to solve. This cuts short the fakers and provides additional needed practice for those who genuinely needed my help.

    (Yeah, I tend to over-analyze things to death, but students will try to find any dodge to get out of doing work if they can--I know since I was the same way when I was their age--so I try to at least make it harder for such "shenanigans" to take place.)

    But, your approach of student's trading journals would lend itself just fine to collecting of errors for later class review.

  14. Okay, I've had a little time to look at your writing prompt files. For what it's worth, this is my suggestion on how to use them.

    1. These are good higher order thinking questions in alignment with Bloom's Taxonomy. A few examples from the files:

    How does an expression differ from an equation? Give an example of each.
    Explain the difference between a radical expression and a rational expression. Give an example of each.
    If we were to factor -3bc-3abc, would the answer be -3bc(-a)? Why or why not?
    Compare three ways of graphing a linear equation.

    2. Start out the semester by asking these types of questions during the lesson and getting oral responses from both volunteers and students you call on. The students who answer are modeling acceptable and unacceptable responses to the questions for the rest of the class.

    3. Gradually introduce you (the teacher) writing a summary at the board of each response.

    4. Gradually introduce asking students how to write the summary and you be scribe at the board.

    5. Gradually, after you've modeled with steps 2-4, have students writing a summary of the volunteer's response in their journal.

    6. Students are asked the question and independently come up with a response and record it in their journal.

    The above is just one approach, and probably not the best. But my thought is that even if you don't get past step 4, you will still be developing their higher order/critical thinking skills. Also, if you try journaling and it becomes too much for some of your students, you don't have to throw in the towel completely: you can always do steps 2,3 and/or 4.

  15. From an e-mail for your viewing pleasure:

    Here is what I tried to post (as gasstationwithoutpumps):

    "It seems to me that if students have weak math skills that writing
    would highlight that, not disguise it. Fortunately, I have all native
    English speakers."

    I wrote a long response yesterday, which blogger promptly discarded,
    so that even the "back" button couldn't retrieve it. This seems to be
    a common problem with Google applications (I have the same trouble
    with Google calendar).

    I'll try again.

    I'm in California, where essentially every school (even through the
    university) has at least 10% and often 50% non-native speakers, so it
    has become reflexive to think about the effect of any assignment on
    non-native speakers.

    I'm teaching at grad school and senior college level, and my students
    have to do a lot of writing, so I do believe that students need to
    learn to write about technical material (not just literary analysis,
    which is all most English teachers use as writing prompts). I even
    created and taught a writing course for engineers for 14 years, so I'm
    not trying to say that teaching students to write is unimportant.

    I do think that writing skills and mathematical skills are not tightly
    coupled, though. Most of the math majors and math grad students I
    went to school with could not write coherent sentences, even when they
    could put together very clean, elegant proofs. Similarly, the students
    who wrote well on other subjects could not construct a complex
    mathematical argument.

    The problem is that "surface" effects such as grammar and diction can
    mask the underlying argument. Bad grammar and diction can make a
    mathematically sound argument appear to be incoherent, and good
    grammar and diction can mask an inherently unsound argument (the
    classic b*s* paper).

    The best way to develop mathematical reasoning is to do problems (not
    drill exercises) that are a little difficult for the individual
    student, requiring real thought to solve. The difficulty for public
    schools is that the students are at a wide range of abilities, and
    what is a boring drill exercise for the top of the class is impossibly
    hard for the bottom of the class.

    I let my son drop his trig class in school (which had loads of boring
    review drill, not even getting to trig for the first month of 30
    problems a night) in order to take the Art of Problem Solving
    Precalculus online class, which only gives 9-10 problems a week, but
    each problem takes some thought to solve. The pace and difficulty of
    the course is a better match for his learning style, but it would not
    be suitable for student who needs smaller steps and more practice to
    pick up new ideas.

  16. Paul,
    I have a shorter way to describe your approach: I do, we do, you do. It's how I do regular notes and examples anyway so yes, I would definitely model journaling in the same manner. I'd also like to have students view student examples so together as a class so we can build a rubric. I want students to clearly know what is expected.

    If the best way to develop mathematical reasoning is to do problems that are a little difficult for the individual
    student, doesn't it stand to reason that writing that solution down after soling it would be a great reinforcement of the process they used. I'm working toward students being able to self-check and self-analyze as well as better retain information. If the problem requires thought, isn't it reasonable to teach students another method of expressing that thought?

    PS Always copy your comment before clicking publish. ;)

  17. Writing down a solution is useful, but that is quite a different exercise from the journaling that you proposed.